Unit 2 : Random Variables and their Distributions

Unit 2 : Random Variables and their Distributions

Wenyaw ChanDivision of BiostatisticsSchool of Public Health

University of Texas- Health Science Center at Houston

Random Variable• Random Variable:

– A numeric function that assigns probabilities to different events in a sample.

• Discrete Random Variable: – A random variable that assumes only a finite or denumerable

number of values. – The probability mass function of a discrete random variable X that

assumes values x1, x2,… is p(x1), p(x2), …., where p(xi)=Pr[X= xi].

• Continuous Random Variable: – A random variable whose possible values cannot be enumerated.

Example: Flip a coin 3 times• Random Variable

– X = # of heads in the 3 coin tosses• Probability Mass Function

– P(X=3) = P{(HHH)} =1/8– P(X=2) = P{HHT, HTH, THH}= 3/8– P(X=1) = P{HTT,THT, TTH} = 3/8– P(X=0) = P{TTT} = 1/8

• X is a discrete random variable with probability (mass) function

x 0 1 2 3

P(X=x) 1/8 3/8 3/8 1/8

Random Variable

Expected value of X :

Variance of X :

Standard Deviation of X: =

k

iii xXxXE

1)Pr()(

2)( XVar

k

iii xXx

1

2 )Pr()(

)(XVar

Random Variable

• Note :

• Cumulative Distribution Function– of X : Pr(X<=x) = F(x)

2)()( XEXVar22 )]([)( XEXE

Binomial Distribution

• Examples of the binomial distribution have a common structure:– n independent trials– each trial has only two possible outcomes, called

“success” and “failure”.– Pr (success) = p for all trials

Binomial Distribution• If X= # of successful trials in these n trials, then X has a

binomial distribution.

• k=0,1,2,….,n • where • Example: Flip a coin 10 times

knkn

kppkXP

)1(

!( )! !

n nk n k k

Properties of Binomial Distribution

• If X~ Binomial (n, p), thenE(X) = np

Var (X) = np(1-p)

Poisson Distribution

k=0,1,2,…..

If X~ Poisson (), then EX = and VarX =

!

Prk

ekXk

Poisson Process• Assumption 1:

– Pr {1 event occurs in a very small time interval [0,t)} t– Pr {0 event occurs in a very small time interval [0, t)}1- t– Pr{more than one event occurs in a very small time interval [0, t)}0

• Assumption 2:– Probability that the number of events occur per unit time is the same

through out the entire time interval • Assumption 3:

– Pr {one event in [t1,t2) | one event in [t0, t1)} = Pr {one event in [t1, t2)}

Poisson Distribution• X=The number of events occurred in the time period t for

the above process with parameter, then mean=t and

where k= 0,1,2,… and e= 2.71828

E(X)=Var(X)=t

!)()(Pr

ktekXkt

Poisson approximation to Binomial

• If X~ Binomial (n, p), n is large and p is small, then

!)()(

knpekXPknp

Continuous Probability Distributions

• Probability density function (p.d.f.) (of a random variable):– a curve such that the area under the curve

between any two points a and b, equals – Prob[a x b ]= ∫ a x bf(x)dx

a b

Pr(a<=X<=b)

Continuous Probability Distributions

• Cumulative distribution function: Pr(x a)

a

Pr(X<=a)

Continuous Probability Distributions

• The expected value of a continuous random variable X is

∫ xf(x)dx, where f(x) is the p.d.f. of X.• The definition for the variance of a

continuous random variable is the same as that of a discrete random variable, i.e.

Var(X)=E(X2)- (EX)2=∫(x-µ)2f(x)dx, where µ=E(X).

The Normal Distribution (The Gaussian distribution)

•

• The p.d.f. of a normal distribution exp , - < x <

21)( xf

2

2 )(2

1

x

The Normal Distribution

• figures: a bell-shaped curve symmetric about

• Notation: X~N(, 2 ) : mean

2 : variance

u-s u u+s

s s

point of inflection

The Normal Distribution

• N(0,1) is the standard normal distribution• If X~ N(0,1), then

– ~ : “is distributed as” , – : c.d.f. for the standard normal r.v.

• Note:– The point of inflection is a point where the slope of

the curve changes its direction.

)Pr()( xXx

Properties of the N(0,1)

• 1. (-x) = 1-(x)• 2.

– About 68% of the area under the standard normal curve lies between –1 and 1.

– About 95% of the area under the standard normal curve lies between –2 and 2.

– About 99% of the area under the standard normal curve lies between –2.5 and 2.5.

Properties of the N(0,1)

• If X~ N(0,1) and P(X< Zu)=u, 0 u 1 then Zu is called the 100uth percentile of the

standard normal distribution.95th %tile=1.645, 97.5th %tile=1.96, 99th %tile=2.33

Zu

Area=u

Properties of the N(0,1)

• If X~ N(, 2), then

• This property allows us to calculate the probability of a non-standard normal random variable.

~ (0,1)X N

r ra X bP a X b P

ab

Other Distributions--- t distribution

• Let X1, ….Xn be a random sample from a normal population N(, σ2).

Then

has a t distribution with n-1 degrees of freedom (df).

/Xs n

Other Distributions---Chi-square distribution

• Let X1, ….Xn be a random sample from a normal population N(0, 1).

Then

has a chi-square distribution with n degrees of freedom (df).

2

1

n

ii

X

Other Distributions---F distribution

• Let U and V be independent random variables and each has a chi-square distribution with p and q degrees of freedom respectively.

Then

has a F distribution with p and q degrees of freedom (df).

//

U pV q

Covariance and Correlation

• The covariance between two random variables is defined by

Cov(X,Y)=E[(X-µX)(Y-µY)].• The correlation coefficient between two

random variables is defined by ρ=Corr(X,Y)=Cov(X,Y)/(σX σ Y).

Variance of a Linear Combination

• Var(c1X1 + c2X2)

),(2

)()(

2121

2221

21

XXCovcc

XVarcXVarc

),(2

)()(

2121

2221

21

XXCorrcc

XVarcXVarc

YX