MAS113 Introduction to Probability and Statistics · Discrete random variables Expectation and variance Standard discrete probability distributions MAS113 Introduction to Probability

Discrete random variablesExpectation and variance

Standard discrete probability distributions

MAS113 Introduction to Probability and

Statistics

Dr Jonathan Jordan

School of Mathematics and Statistics, University of Sheffield

2017–18

Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics



Random variables

Informally, we think of a random variable as any quantitythat is uncertain to us. For example:

the number of emergency call-outs received by a firestation in a given week;

the price of a barrel of oil in one month’s time;

the number of gold medals won by Great Britain at thenext summer Olympics.




Random variables








Random variables








Random variables








Random variables continued

We cannot say with certainty what any of these quantities are,but probability theory gives us a framework for describing howlikely different values are.

Whereas elements of a sample space may not be numerical,random variables are always numerical quantities, and so,when defining a random variable, we need a rule for gettingfrom the random outcome in the sample space to the value ofthe random variable.





We cannot say with certainty what any of these quantities are,but probability theory gives us a framework for describing howlikely different values are.

Whereas elements of a sample space may not be numerical,random variables are always numerical quantities, and so,when defining a random variable, we need a rule for gettingfrom the random outcome in the sample space to the value ofthe random variable.





Definition

Given a sample space S , we define a random variable X tobe a mapping from S to the real line R.

We sometimes write a random variable as X (s), where s ∈ S .We define the range of X to be the set of all possible valuesof X :

RX := {x ∈ R; x = X (s) for some s ∈ S}.





Definition

Given a sample space S , we define a random variable X tobe a mapping from S to the real line R.We sometimes write a random variable as X (s), where s ∈ S .

We define the range of X to be the set of all possible valuesof X :






Definition

Given a sample space S , we define a random variable X tobe a mapping from S to the real line R.We sometimes write a random variable as X (s), where s ∈ S .We define the range of X to be the set of all possible valuesof X :





Discrete random variables

In this chapter, we consider discrete random variables, inwhich the number of possible values is either finite orcountably infinite.

Example

Counting heads in coin tosses

Example

Share portfolio






Example


Example

Share portfolio






Example


Example

Share portfolio




Summation notation

Before continuing with the discussion of random variables, wedefine some new summation notation, and recap some resultsregarding manipulations of sums.

Let X be a discrete random variable with rangeRX = {x1, x2, . . . , xn}.

For any function g(x), we define

∑x∈Rx

g(x) :=n∑

i=1

g(xi) = g(x1) + g(x2) + . . . + g(xn).




Summation notation




∑x∈Rx

g(x) :=n∑

i=1

g(xi) = g(x1) + g(x2) + . . . + g(xn).




Summation notation




∑x∈Rx

g(x) :=n∑

i=1

g(xi) = g(x1) + g(x2) + . . . + g(xn).




Summation notation continued

For any two constants a and b, we have

n∑i=1

(a + bg(xi)) = (a + bg(x1)) + . . . + (a + bg(xn))

= na + bn∑

i=1

g(xi).

Note thatn∑

i=1

a = na,

(so the sum is not equal to a).






n∑i=1

(a + bg(xi)) = (a + bg(x1)) + . . . + (a + bg(xn))

= na + bn∑

i=1

g(xi).

Note thatn∑

i=1

a = na,







n∑i=1

(a + bg(xi)) = (a + bg(x1)) + . . . + (a + bg(xn))

= na + bn∑

i=1

g(xi).

Note thatn∑

i=1

a = na,







n∑i=1

(a + bg(xi)) = (a + bg(x1)) + . . . + (a + bg(xn))

= na + bn∑

i=1

g(xi).

Note thatn∑

i=1

a = na,







n∑i=1

(a + bg(xi)) = (a + bg(x1)) + . . . + (a + bg(xn))

= na + bn∑

i=1

g(xi).

Note thatn∑

i=1

a = na,





Probability mass functions

Definition

For a discrete random variable X , we define the probabilitymass function (p.m.f. for short) pX to be

pX (x) := P(X = x),

where x can be any real number.Note that P(X = x) = P(A) where

A = {s ∈ S ; X (s) = x}.





Definition


pX (x) := P(X = x),


A = {s ∈ S ; X (s) = x}.





Definition


pX (x) := P(X = x),

where x can be any real number.

Note that P(X = x) = P(A) where

A = {s ∈ S ; X (s) = x}.





Definition


pX (x) := P(X = x),


A = {s ∈ S ; X (s) = x}.




Notation

In the notation, we use X to represent the random variable,and x to represent a possible value of X

Whereas X refers to a specific random variable, the use of theletter x is arbitrary; we could just as well writepX (a) := P(X = a).




Notation

In the notation, we use X to represent the random variable,and x to represent a possible value of X

Whereas X refers to a specific random variable, the use of theletter x is arbitrary; we could just as well writepX (a) := P(X = a).




Properties

A probability mass function must have the following twoproperties.

1 pX (x) ≥ 0∀x ∈ R.

2 Probability mass functions must ‘sum to 1’:∑x∈RX

pX (x) = 1.




Properties


1 pX (x) ≥ 0∀x ∈ R.


pX (x) = 1.




Properties


1 pX (x) ≥ 0∀x ∈ R.


pX (x) = 1.




Proofs of properties

Property 1 follows from the definition of pX (x).

To prove property 2, first write RX = {x1, x2, . . . , xn}, and letAi = {s ∈ S ; X (s) = xi}.

You should now convince yourself that A1, . . . ,An is apartition of S .


















Example

Example

In a simple lottery, two numbers are drawn at random, withoutreplacement, from the numbers 1,2,3,4.You choose two numbers: 1 and 3.

If both 1 and 3 are drawn, you win £10. If either 1 or 3 isdrawn (but not both), you win £5. Otherwise, you winnothing.Let X be the amount in pounds that you win. Tabulate pX (x).




Example

Example

In a simple lottery, two numbers are drawn at random, withoutreplacement, from the numbers 1,2,3,4.You choose two numbers: 1 and 3.If both 1 and 3 are drawn, you win £10. If either 1 or 3 isdrawn (but not both), you win £5. Otherwise, you winnothing.

Let X be the amount in pounds that you win. Tabulate pX (x).




Example

Example

In a simple lottery, two numbers are drawn at random, withoutreplacement, from the numbers 1,2,3,4.You choose two numbers: 1 and 3.If both 1 and 3 are drawn, you win £10. If either 1 or 3 isdrawn (but not both), you win £5. Otherwise, you winnothing.Let X be the amount in pounds that you win. Tabulate pX (x).




Measure

Note that a probability mass function can be thought of asdefining a probability measure on the range space RX .

For a subset A ⊆ RX , define mX (A) := P(X ∈ A).

It is not hard to check that mX satisfies the definition of aprobability measure, and it is called the law or distribution ofX .

We generally think in terms of the probability mass functionrather than of the measure, but the measure idea is usefulwhen we come to generalise beyond discrete random variables.




Measure








Measure








Measure








Cumulative distribution function

Definition

We define the cumulative distribution function,abbreviated to c.d.f., FX of a random variable X to be

FX (x) := P(X ≤ x),






Definition


FX (x) := P(X ≤ x),






Definition


FX (x) := P(X ≤ x),





C.d.f. and p.d.f.

The cumulative distribution function can be written in termsof the probability mass function:

FX (x) := P(X ≤ x) =∑

a≤x ,a∈RX

pX (a). (1)




C.d.f. and p.d.f.

The cumulative distribution function can be written in termsof the probability mass function:

FX (x) := P(X ≤ x) =∑

a≤x ,a∈RX

pX (a). (1)




Example

Example

Suppose England are to play the West Indies in a 3 match testseries. Let X be the number of matches won by England. Ifmy probability mass function for X is

pX (0) = 0.05, pX (1) = 0.2, pX (2) = 0.6, pX (3) = 0.15,

tabulate my cumulative distribution function.




Quantile function

The quantile function is related to the inverse of thecumulative distribution function.

Definition

For α ∈ [0, 1] the α quantile (or 100× α percentile) is thesmallest value of x such that

FX (x) ≥ α

The median is the 0.5 quantile.




Quantile function


Definition


FX (x) ≥ α





Quantile function


Definition


FX (x) ≥ α





Independence of random variables

We say that two random variables X and Y are independentof each other if any event defined only using the value of X isindependent of any event defined only using the value of Y .

More specifically, we can make the following definition fordiscrete random variables:

Definition

Two discrete random variables X and Y are independent if

P(X = x ,Y = y) = P(X = x)P(Y = y),

for all x and y , or, equivalently, . . .





We say that two random variables X and Y are independentof each other if any event defined only using the value of X isindependent of any event defined only using the value of Y .

More specifically, we can make the following definition fordiscrete random variables:

Definition

Two discrete random variables X and Y are independent if

P(X = x ,Y = y) = P(X = x)P(Y = y),

for all x and y , or, equivalently, . . .




Independence continued

Definition. . .

P(X = x |Y = y) = P(X = x).

If two random variables are not independent, then we say thatthey are dependent.




Example

Example





Expectation and variance

The probability mass function gives a complete description ofthe uncertainty we have about a random variable X .

It tells us how likely each possible value of X is.

However, there are other quantities that can tell us usefulthings about a random variable, which we can derive from theprobability mass function. We consider here the expectationand variance of a random variable.


















Expectation I

On a European roulette wheel, the ball can land on one of theintegers 0 to 36.

A bet of one pound on odd returns one pound (plus theoriginal stake) if the ball lands on any odd number from 1 to35.

Assuming the ball is equally likely to land anywhere, if you betone pound on odd a large number of times, how much moneyper game are you likely to win (or lose)?




Expectation I







Expectation I







Expectation II

Informally, if you bet on odd 10000 times, we might supposethat you will win 18

37× 10000 = 4865 times and lose

1937× 10000 = 5135 times, so you will lose 270 pounds overall,

or 2.7 pence per game.

Formally, we define the expected profit (or loss) per game.




Expectation

Definition

The expectation E (X ) of a discrete random variable X isdefined as

E (X ) :=∑x∈RX

xP(X = x)

We refer to the expectation of X as the mean of X and writeµX to represent the mean:

µX := E (X ).




Expectation

Definition

The expectation E (X ) of a discrete random variable X isdefined as

E (X ) :=∑x∈RX

xP(X = x)

We refer to the expectation of X as the mean of X and writeµX to represent the mean:

µX := E (X ).




Example

Example

In the roulette example, let X be your net winnings after asingle bet of one pound on odd. What is E (X )?




Example

Example

Suppose Everton are to play Chelsea in the Premier League,and that you believe fair odds against Everton winning are 3to 1, so that your probability that Everton win is 0.25.

If you offer these odds to someone, and they place a £1 bet,what is your expected profit?Suppose a bookmaker also judges the probability that Evertonwin is 0.25, but instead offers odds of 12 to 5 against. Ifsomeone places a £1 bet on Everton winning, what is thebookmaker’s expected profit?




Example

Example

Suppose Everton are to play Chelsea in the Premier League,and that you believe fair odds against Everton winning are 3to 1, so that your probability that Everton win is 0.25.If you offer these odds to someone, and they place a £1 bet,what is your expected profit?

Suppose a bookmaker also judges the probability that Evertonwin is 0.25, but instead offers odds of 12 to 5 against. Ifsomeone places a £1 bet on Everton winning, what is thebookmaker’s expected profit?




Example

Example

Suppose Everton are to play Chelsea in the Premier League,and that you believe fair odds against Everton winning are 3to 1, so that your probability that Everton win is 0.25.If you offer these odds to someone, and they place a £1 bet,what is your expected profit?Suppose a bookmaker also judges the probability that Evertonwin is 0.25, but instead offers odds of 12 to 5 against. Ifsomeone places a £1 bet on Everton winning, what is thebookmaker’s expected profit?




Example

Example

Let X be a random variable with RX = {−1, 0, 1}.

Define Y = g(X ) = X 2.Then Y is another random variable, with RY (y) = {0, 1}.What is E{g(X )}?




Example

Example

Let X be a random variable with RX = {−1, 0, 1}.Define Y = g(X ) = X 2.

Then Y is another random variable, with RY (y) = {0, 1}.What is E{g(X )}?




Example

Example

Let X be a random variable with RX = {−1, 0, 1}.Define Y = g(X ) = X 2.Then Y is another random variable, with RY (y) = {0, 1}.What is E{g(X )}?




Expectation of a function of X

Sometimes it is useful to calculate the expectation of afunction of X . The following result generalises the previousexample to tell us how.

Theorem

(The expectation of g(X ))For any function g of a random variable X , with probabilitymass function pX (x),

E{g(X )} =∑x∈RX

g(x)pX (x).






Theorem


E{g(X )} =∑x∈RX

g(x)pX (x).






Theorem


E{g(X )} =∑x∈RX

g(x)pX (x).




Variance

If we can repeat the experiment and observe X lots of times,informally, the expectation of X tells us what we are likely tosee ‘on average’.

(We will consider this more carefully when we study sums ofrandom variables).

It will also be useful to consider how far X might be from itsexpectation.




Variance







Variance







Distance from mean

Consider two random variables X and Y , with the followingprobability mass functions:

pX (32) = 13, pX (36) = 1

3, pX (46) = 1

3,

pY (12) = 13, pY (20) = 1

3, pY (82) = 1

3.

Then

E (X ) = 32× 1

3+ 36× 1

3+ 46× 1

3= 38,

E (Y ) = 12× 1

3+ 20× 1

3+ 82× 1

3= 38.

Both X and Y have the same expected value, but forwhatever values of X and Y we observe, X will be closer toE (X ) than Y will be to E (Y ).




Distance from mean


pX (32) = 13, pX (36) = 1

3, pX (46) = 1

3,

pY (12) = 13, pY (20) = 1

3, pY (82) = 1

3.

Then

E (X ) = 32× 1

3+ 36× 1

3+ 46× 1

3= 38,

E (Y ) = 12× 1

3+ 20× 1

3+ 82× 1

3= 38.





Distance from mean


pX (32) = 13, pX (36) = 1

3, pX (46) = 1

3,

pY (12) = 13, pY (20) = 1

3, pY (82) = 1

3.

Then

E (X ) = 32× 1

3+ 36× 1

3+ 46× 1

3= 38,

E (Y ) = 12× 1

3+ 20× 1

3+ 82× 1

3= 38.





Distance from mean


pX (32) = 13, pX (36) = 1

3, pX (46) = 1

3,

pY (12) = 13, pY (20) = 1

3, pY (82) = 1

3.

Then

E (X ) = 32× 1

3+ 36× 1

3+ 46× 1

3= 38,

E (Y ) = 12× 1

3+ 20× 1

3+ 82× 1

3= 38.





Distance from mean


pX (32) = 13, pX (36) = 1

3, pX (46) = 1

3,

pY (12) = 13, pY (20) = 1

3, pY (82) = 1

3.

Then

E (X ) = 32× 1

3+ 36× 1

3+ 46× 1

3= 38,

E (Y ) = 12× 1

3+ 20× 1

3+ 82× 1

3= 38.





Definition

We use the concept of variance to describe how close arandom variable is likely to be to its expected value.

Definition

The variance Var(X ) of a discrete random variable X isdefined as

Var(X ) : = E[{X − E (X )}2

]= E{(X − µX )2}

=∑x∈RX

(x − µX )2pX (x).

We denote the variance by σ2X :

σ2X := Var(X ).




Definition


Definition


Var(X ) : = E[{X − E (X )}2

]= E{(X − µX )2}

=∑x∈RX

(x − µX )2pX (x).


σ2X := Var(X ).




Definition


Definition


Var(X ) : = E[{X − E (X )}2

]= E{(X − µX )2}

=∑x∈RX

(x − µX )2pX (x).


σ2X := Var(X ).




Definition


Definition


Var(X ) : = E[{X − E (X )}2

]= E{(X − µX )2}

=∑x∈RX

(x − µX )2pX (x).


σ2X := Var(X ).




Why squared?

If you are wondering why the variance is defined asE [{X − E (X )}2] rather than E{X − E (X )}, the latterexpression will not tell us anything useful about X :

Theorem

(The expected difference between a random variable and itsmean)

E{X − E (X )} = 0,

for any random variable X .




Why squared?

If you are wondering why the variance is defined asE [{X − E (X )}2] rather than E{X − E (X )}, the latterexpression will not tell us anything useful about X :

Theorem

(The expected difference between a random variable and itsmean)

E{X − E (X )} = 0,

for any random variable X .




Standard deviation

As the variance is defined as an expected squared difference,the variance will be expressed in units that are the square ofthe units of X .

If we want a measure of spread that is in the same units as X ,we take the square root of the variance.

Definition

The standard deviation of a random variable X , denoted byσX , is the square root of the variance of X .

σX :=√

Var(X ).




Standard deviation



Definition


σX :=√

Var(X ).




Standard deviation



Definition


σX :=√

Var(X ).




The variance identity

The following result is useful for calculating variances:

Theorem

Var(X ) = E (X 2)− E (X )2,




The variance identity

The following result is useful for calculating variances:

Theorem

Var(X ) = E (X 2)− E (X )2,




Calculating the variance

To calculate a variance, if we already have E (X ), we just needto calculate E (X 2).

(Alternatively, if we know the mean and variance, this gives usa quick way of calculating E (X 2)).

Note that as long as Var(X ) > 0 we can see that

E (X 2) 6= E (X )2.








E (X 2) 6= E (X )2.








E (X 2) 6= E (X )2.




Example

Example

Let X be the random variable defined in the roulette example,with E (X ) = −1/37. What is Var(X )?




The expectation and variance of aX + b

Theorem

Let X be a random variable, and a and b be any twoconstants. Then

E (aX + b) = aE (X ) + b,

Var(aX + b) = a2 Var(X ).




The expectation and variance of aX + b

Theorem

Let X be a random variable, and a and b be any twoconstants. Then

E (aX + b) = aE (X ) + b,

Var(aX + b) = a2 Var(X ).




Special cases

If we set a = 0, then we can see that for any constant b,

E (b) = b,

Var(b) = 0.




Expectation of a sum

It is often useful to consider expectations of sums of randomvariables:

Theorem

Given any two random variables X and Y

E (X + Y ) = E (X ) + E (Y ).




Expectation of a product

We might hope that the same was true for variance, so thatthe variance of X + Y was the sum of the variances of X andY .

This is not true in general, but it is true when X and Y areindependent. To prove this we will first of all prove animportant result about the expectation of a product ofindependent random variables.

Theorem

For any two random variables X and Y which are independent,

E (XY ) = E (X )E (Y ).







Theorem


E (XY ) = E (X )E (Y ).







Theorem


E (XY ) = E (X )E (Y ).




Variance of independent sum

Corollary


Var(X + Y ) = Var(X ) + Var(Y ).




Example

Example

Let X and Y be independent random variables withVar(X ) = 9 and Var(Y ) = 16. What is Var(X − Y )?




Standard probability distributions

We now consider some standard probability distributions fordiscrete random variables, that can be used in a variety ofdifferent applications.

By “distribution”, we mean a particular choice of probabilitymass function, which may be specified in terms of someparameters.




Standard probability distributions

We now consider some standard probability distributions fordiscrete random variables, that can be used in a variety ofdifferent applications.

By “distribution”, we mean a particular choice of probabilitymass function, which may be specified in terms of someparameters.




The Bernoulli distribution

A Bernoulli random variable X can take one of two values: 0and 1.

Examples of ‘experiments’ that we might describe using aBernoulli random variable are

a patient is given a drug, and the drug either ‘works’:X = 1, or does not: X = 0;

a tennis player either wins a match: X = 1, or loses:X = 0;

in one year, a house is either burgled: X = 1, or not:X = 0.








































Definition

Definition

If a random variable X has a Bernoulli distribution, then itsprobability mass function is

pX (1) = p,

pX (0) = 1− p,

and pX (x) = 0 otherwise, with 0 ≤ p ≤ 1.We write

X ∼ Bernoulli(p),

to mean “X has a Bernoulli distribution with parameter p’(the probability that X = 1)”.




Definition

Definition


pX (1) = p,

pX (0) = 1− p,


X ∼ Bernoulli(p),





Definition

Definition


pX (1) = p,

pX (0) = 1− p,


X ∼ Bernoulli(p),





Definition

Definition


pX (1) = p,

pX (0) = 1− p,

and pX (x) = 0 otherwise, with 0 ≤ p ≤ 1.

We writeX ∼ Bernoulli(p),





Definition

Definition


pX (1) = p,

pX (0) = 1− p,


X ∼ Bernoulli(p),





Definition

Definition


pX (1) = p,

pX (0) = 1− p,


X ∼ Bernoulli(p),





Mean and variance

Theorem

(Expectation and variance of a Bernoulli random variable)For the expectation of a Bernoulli random variableX ∼ Bernoulli(p), we have

E (X ) = p,

Var(X ) = p(1− p).




Mean and variance

Theorem

(Expectation and variance of a Bernoulli random variable)For the expectation of a Bernoulli random variableX ∼ Bernoulli(p), we have

E (X ) = p,

Var(X ) = p(1− p).




The binomial distribution

Consider the following situations:

100 patients are given a drug. Each patient either‘responds’ to the drug, or does not. X is the number ofpatients that respond to the drug.

in a crime survey, 1000 people are selected at random,and asked whether they have been burgled in the lastyear. X is the number of people who respond ‘yes’.

in a quality control procedure, 20 items are selected atrandom, and tested to see whether they are faulty. X isthe number of faulty items.




















Bernoulli trials

In each case we have a fixed number of “trials”, each of whichcan have two possible outcomes (often called “success” and“failure”).

(Each trial can be considered an example of a Bernoullidistribution, with “success” corresponding to 1 and “failure”to 0, so they are often referred to as Bernoulli trials.)

In each of these situations it is reasonable to assume that theprobability of a “success”, which we will call p, is constantfrom from one trial to the next, and that the trials areindependent.

In each case we are counting the total number of successes.




Bernoulli trials








Bernoulli trials








Bernoulli trials




In each case we are counting the total number of successes.Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics



n = 2

To think about the form of the probability mass function of X ,consider the case n = 2. The possible outcomes are

(on board)

Consider calculating pX (1).

We are not interested in which trials are successes, only thetotal number of successes

There are two ways of achieving one success in total, and foreach of these possibilities, the corresponding probability isp(1− p), so we have pX (1) = 2p(1− p).




n = 2


(on board)







n = 2


(on board)







General formula

In general for n trials, the number of possible sequences thatcontain x successes in total will be

(n

x

)=

n!

x!(n − x)!,

and the probability of any individual sequence with x successesin total will be px(1− p)n−x .

So we will have pX (x) =(nx

)px(1− p)n−x .




General formula

In general for n trials, the number of possible sequences thatcontain x successes in total will be(

n

x

)=

n!

x!(n − x)!,



)px(1− p)n−x .




General formula


n

x

)=

n!

x!(n − x)!,



)px(1− p)n−x .




General formula


n

x

)=

n!

x!(n − x)!,



)px(1− p)n−x .




Definition

Motivated by this, we make the following definition:

Definition

If a random variable X has a binomial distribution, withparameters n (the number of trials) and p (the probability ofsuccess in each trial), then the probability mass function of Xis given by

pX (x) =n!

x!(n − x)!px(1− p)n−x ,

for x ∈ RX = {0, 1, 2, . . . , n}, and 0 otherwise.We write X ∼ Bin(n, p), to mean “X has a binomialdistribution with parameters n (the number of trials) and p(the probability of success in each trial)”.




Definition


Definition


pX (x) =n!

x!(n − x)!px(1− p)n−x ,





Definition


Definition


pX (x) =n!

x!(n − x)!px(1− p)n−x ,





Definition


Definition


pX (x) =n!

x!(n − x)!px(1− p)n−x ,

for x ∈ RX = {0, 1, 2, . . . , n}, and 0 otherwise.

We write X ∼ Bin(n, p), to mean “X has a binomialdistribution with parameters n (the number of trials) and p(the probability of success in each trial)”.




Definition


Definition


pX (x) =n!

x!(n − x)!px(1− p)n−x ,

for x ∈ RX = {0, 1, 2, . . . , n}, and 0 otherwise.We write X ∼ Bin(n, p),

to mean “X has a binomialdistribution with parameters n (the number of trials) and p(the probability of success in each trial)”.




Definition


Definition


pX (x) =n!

x!(n − x)!px(1− p)n−x ,





Link to binomial theorem

Note that the binomial theorem confirms that the binomialprobability mass function is valid (ie it sums to 1):

It tells us that

(a + b)n =n∑

x=0

(n

x

)axbn−x .

and if we now choose a = p and b = 1− p, we have

n∑x=0

pX (x) =n∑

x=0

(n

x

)px(1− p)n−x = (p + (1− p))n = 1,

as we should have.






It tells us that

(a + b)n =n∑

x=0

(n

x

)axbn−x .


n∑x=0

pX (x) =n∑

x=0

(n

x

)px(1− p)n−x = (p + (1− p))n = 1,

as we should have.






It tells us that

(a + b)n =n∑

x=0

(n

x

)axbn−x .


n∑x=0

pX (x) =n∑

x=0

(n

x

)px(1− p)n−x = (p + (1− p))n = 1,

as we should have.






It tells us that

(a + b)n =n∑

x=0

(n

x

)axbn−x .


n∑x=0

pX (x) =n∑

x=0

(n

x

)px(1− p)n−x = (p + (1− p))n = 1,

as we should have.Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics



Mean and variance

Theorem

(Expectation and variance of a binomial random variable)For X ∼ Bin(n, p) we have

E (X ) = np

Var(X ) = np(1− p).




Mean and variance

Theorem

(Expectation and variance of a binomial random variable)For X ∼ Bin(n, p) we have

E (X ) = np

Var(X ) = np(1− p).




Proportion of successes

As well as being interested in the total number of ‘successes’X , we may also be interested in the proportion of success X/n.

We have

E

(X

n

)= p,

Var

(X

n

)=

p(1− p)

n.

What do you think will happen to X/n as n→∞?






We have

E

(X

n

)= p,

Var

(X

n

)=

p(1− p)

n.







We have

E

(X

n

)= p,

Var

(X

n

)=

p(1− p)

n.







We have

E

(X

n

)= p,

Var

(X

n

)=

p(1− p)

n.





The cumulative distribution function

The cumulative distribution function is given by

FX (x) = P(X ≤ x) =x∑

a=0

pX (a)

=x∑

a=0

n!

a!(n − a)!pa(1− p)n−a.

We cannot simplify this expression, and so calculating thec.d.f. by hand can be tedious.

Fortunately, we can do this and other calculations related tothe binomial distribution very easily in R.






FX (x) = P(X ≤ x) =x∑

a=0

pX (a)

=x∑

a=0

n!

a!(n − a)!pa(1− p)n−a.








FX (x) = P(X ≤ x) =x∑

a=0

pX (a)

=x∑

a=0

n!

a!(n − a)!pa(1− p)n−a.








FX (x) = P(X ≤ x) =x∑

a=0

pX (a)

=x∑

a=0

n!

a!(n − a)!pa(1− p)n−a.






The binomial distribution in R

As with most standard distributions in R, there are commandsfor calculating the p.m.f., c.d.f., quantile function, and forrandomly sampling from the distribution.




The binomial distribution: examples

Example

A company claims that, for a particular product, 8 out of 10people prefer their brand A over a rival’s brand B .

You randomly sample 50 people, and ask them whether theyprefer brand A to brand B .Let X be the number of people who choose brand A.





Example

A company claims that, for a particular product, 8 out of 10people prefer their brand A over a rival’s brand B .You randomly sample 50 people, and ask them whether theyprefer brand A to brand B .

Let X be the number of people who choose brand A.





Example

A company claims that, for a particular product, 8 out of 10people prefer their brand A over a rival’s brand B .You randomly sample 50 people, and ask them whether theyprefer brand A to brand B .Let X be the number of people who choose brand A.





Example

If the company is right,

1 What are the expectation and variance of X ?

2 What is the probability that X = 40?

3 What is the probability that X ≤ 30?

4 What is the probability that X ≥ 45?




The binomial distribution: examplesExample

A study by Burn et al (2011) investigated whether regularlytaking aspirin could reduce the risk of colorectal cancer incarriers of Lynch syndrome.

In a control group, 434 participants regularly took a placebo,and over the period of the study, 30 members of the controlgroup developed primary colorectal cancers.In the treatment group, 427 participants regularly took aspirin,and 18 members developed primary colorectal cancers.Let X be the number of participants in the treatment groupwho developed primary colorectal cancers.Suppose the probability of a participant developing a cancerwas 30/434, regardless of which group the participant was in.





A study by Burn et al (2011) investigated whether regularlytaking aspirin could reduce the risk of colorectal cancer incarriers of Lynch syndrome.In a control group, 434 participants regularly took a placebo,and over the period of the study, 30 members of the controlgroup developed primary colorectal cancers.

In the treatment group, 427 participants regularly took aspirin,and 18 members developed primary colorectal cancers.Let X be the number of participants in the treatment groupwho developed primary colorectal cancers.Suppose the probability of a participant developing a cancerwas 30/434, regardless of which group the participant was in.





A study by Burn et al (2011) investigated whether regularlytaking aspirin could reduce the risk of colorectal cancer incarriers of Lynch syndrome.In a control group, 434 participants regularly took a placebo,and over the period of the study, 30 members of the controlgroup developed primary colorectal cancers.In the treatment group, 427 participants regularly took aspirin,and 18 members developed primary colorectal cancers.

Let X be the number of participants in the treatment groupwho developed primary colorectal cancers.Suppose the probability of a participant developing a cancerwas 30/434, regardless of which group the participant was in.





A study by Burn et al (2011) investigated whether regularlytaking aspirin could reduce the risk of colorectal cancer incarriers of Lynch syndrome.In a control group, 434 participants regularly took a placebo,and over the period of the study, 30 members of the controlgroup developed primary colorectal cancers.In the treatment group, 427 participants regularly took aspirin,and 18 members developed primary colorectal cancers.Let X be the number of participants in the treatment groupwho developed primary colorectal cancers.

Suppose the probability of a participant developing a cancerwas 30/434, regardless of which group the participant was in.





A study by Burn et al (2011) investigated whether regularlytaking aspirin could reduce the risk of colorectal cancer incarriers of Lynch syndrome.In a control group, 434 participants regularly took a placebo,and over the period of the study, 30 members of the controlgroup developed primary colorectal cancers.In the treatment group, 427 participants regularly took aspirin,and 18 members developed primary colorectal cancers.Let X be the number of participants in the treatment groupwho developed primary colorectal cancers.Suppose the probability of a participant developing a cancerwas 30/434, regardless of which group the participant was in.





Example

1 Calculate the probability that no more than 18participants in the treatment group develop primarycolorectal cancers.

2 Find the 2.5th and 97.5th percentiles of X .

Note that formal methods for testing whether there is a“significant” difference between the two groups will be coveredin Semester 2.





Example

1 Calculate the probability that no more than 18participants in the treatment group develop primarycolorectal cancers.

2 Find the 2.5th and 97.5th percentiles of X .

Note that formal methods for testing whether there is a“significant” difference between the two groups will be coveredin Semester 2.




The Poisson distribution

The Poisson distribution is used to represent count data: thenumber of times an event occurs in some finite interval in timeor space.

Some situations that we might model using a Poissondistribution are as follows.

The number of arrivals at an Accident & Emergency wardin one night;

the number of burglaries in a city in a year;

the number of goals scored by a team in a football match;

the number of leaks in 1km section of water pipe.












































Motivation

We can motivate the form of the Poisson distribution asfollows.

Consider the third example, assume that we expect the teamto score about λ goals (for some λ) and imagine dividing thematch up into n short time intervals, where n is some arbitrarybut large number.

(E.g. n could be 90 and each of the intervals one minute long.)




Motivation







Motivation







Motivation continued

Assume that in each of these time intervals the probability ofthe team scoring one goal is p, independently of the otherintervals, and that the probability of them scoring more thanone goal in an interval is “negligible”.

Under these assumptions, the number of goals scored has aBin(n, p) distribution.

That the expectation of a Bin(n, p) random variable is npsuggests that we should now take p = λ/n.

Because we set n to be large, this suggests we should considerthe behaviour of Bin(n, λ/n) for large n.




























Limit of Binomials

Theorem

Consider X ∼ Bin(n, λ/n), and suppose that n→∞.

Then, as n→∞

pX (x)→ e−λλx

x!.




Limit of Binomials

Theorem

Consider X ∼ Bin(n, λ/n), and suppose that n→∞.Then, as n→∞

pX (x)→ e−λλx

x!.




Limit of Binomials

Theorem

Consider X ∼ Bin(n, λ/n), and suppose that n→∞.Then, as n→∞

pX (x)→ e−λλx

x!.




Definition

Motivated by this result, we make the following definition:

Definition

If a random variable X has a Poisson distribution, withparameter λ > 0, then its probability mass function is given by

pX (x) = P(X = x) =e−λλx

x!,

for x ∈ N0 and 0 otherwise.We write X ∼ Poisson(λ),to mean “X has a Poisson distribution with rate parameter λ”.




Definition


Definition



x!,





Definition


Definition



x!,

for x ∈ N0 and 0 otherwise.

We write X ∼ Poisson(λ),to mean “X has a Poisson distribution with rate parameter λ”.




Definition


Definition



x!,





Parameter

The Poisson distribution has a single parameter λ, known asthe rate parameter.

Shortly, we will show that E (X ) = λ, so you can interpret λ asthe expected number of times the event will occur.




Parameter

The Poisson distribution has a single parameter λ, known asthe rate parameter.

Shortly, we will show that E (X ) = λ, so you can interpret λ asthe expected number of times the event will occur.




Sum to 1, mean and variance

Theorem

(Poisson random variable: valid p.m.f., expectation andvariance)

1 The Poisson probability mass function is a validprobability mass function.

2 If X ∼ Poisson(λ) then

E (X ) = Var(X ) = λ.




Sum to 1, mean and variance

Theorem

(Poisson random variable: valid p.m.f., expectation andvariance)

1 The Poisson probability mass function is a validprobability mass function.

2 If X ∼ Poisson(λ) then

E (X ) = Var(X ) = λ.






FX (x) = P(X ≤ x) =x∑

a=0

pX (a)

=x∑

a=0

e−λλa

a!.

As with the binomial distribution, this is tedious to calculateby hand, but easy to calculate using R.






FX (x) = P(X ≤ x) =x∑

a=0

pX (a)

=x∑

a=0

e−λλa

a!.







FX (x) = P(X ≤ x) =x∑

a=0

pX (a)

=x∑

a=0

e−λλa

a!.





The Poisson distribution in R

dpois(x,lambda) for p.m.f.

pppois(x,lambda) for c.d.f.

qpois(alpha,lambda) for quantile function

rpois(m,lambda) for m random observations




The Poisson distribution: examples

Example

Suppose X , the number of accidents at a road junction in oneyear has a Poisson distribution with rate parameter 5.








The Poisson distribution: examples

Example

Suppose X , the number of accidents at a road junction in oneyear has a Poisson distribution with rate parameter 5.








Approximating the Binomial

Note that Theorem 14 suggests that for large n and small p,the Bin(n, p) distribution can be well approximated by thePoisson(np) distribution.

Example

An article published on the BBC news website reported“three-fold variation” in UK bowel cancer death rates. Theaverage death rate from bowel cancer across the UK isreported as 17.6 per 100,000.By considering 100 ‘regions’ each with the same populationsize of 100,000, how much variation could be due to chancealone?






Example

An article published on the BBC news website reported“three-fold variation” in UK bowel cancer death rates. Theaverage death rate from bowel cancer across the UK isreported as 17.6 per 100,000.

By considering 100 ‘regions’ each with the same populationsize of 100,000, how much variation could be due to chancealone?






Example

An article published on the BBC news website reported“three-fold variation” in UK bowel cancer death rates. Theaverage death rate from bowel cancer across the UK isreported as 17.6 per 100,000.By considering 100 ‘regions’ each with the same populationsize of 100,000, how much variation could be due to chancealone?




The geometric distribution


Throwing darts at a dartboard until the bull’s eye is hit;

Buying a national lottery ticket each week until thejackpot is won.

As with the Binomial examples, these involve a sequence ofBernoulli trials, each of which a ‘success’, with probability p,or a ‘failure’, with probability 1− p, but now there is no fixednumber of trials; rather, we repeat the trials until we obtain asuccess.




























Definition

A geometric random variable X is the number of the trial inwhich the first success is observed.

Definition

If a random variable X has a geometric distribution, withparameter p (the probability of a success in any single trial),then the probability mass function of X is given by

pX (x) = (1− p)x−1p,

for x ∈ N and 0 otherwise.We write X ∼ Geometric(p).




Definition


Definition


pX (x) = (1− p)x−1p,





Definition


Definition


pX (x) = (1− p)x−1p,





Definition


Definition


pX (x) = (1− p)x−1p,

for x ∈ N and 0 otherwise.

We write X ∼ Geometric(p).




Definition


Definition


pX (x) = (1− p)x−1p,





C.D.F

Theorem

(Cumulative distribution function of a geometric randomvariable)If X ∼ Geometric(p) and x is a non-negative integer then

FX (x) = 1− (1− p)x .




Valid distribution

Note that as x →∞, (1− p)x → 0, so that

∞∑x=1

pX (x) = FX (∞) = 1,

confirming that the probability mass function is valid.




Mean and variance

Theorem

(Expectation and variance of a geometric random variable)If X ∼ Geometric(p) then

E (X ) =1

p

Var(X ) =1− p

p2.




Mean and variance

Theorem

(Expectation and variance of a geometric random variable)If X ∼ Geometric(p) then

E (X ) =1

p

Var(X ) =1− p

p2.




Example

Example

In section 5.1, we calculated the probability of winning theNational Lottery jackpot with a single ticket is p = 1

45057474.

Suppose I buy one ticket per week. Let X be the week numberin which I first win the jackpot.


2 What is the probability that I don’t win at any time in thenext 50 years?




Example

Example


45057474.







Example

Example


45057474.





MAS113 Introduction to Probability and Statistics · Discrete random variables Expectation and variance Standard discrete probability distributions MAS113 Introduction to Probability

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