Stats chapter 7

Chapter 7

Random Variables

7.1 DISCRETE AND CONTINUOUS RANDOM VARIABLES

Random Variables

A RANDOM VARIABLE is a variable whose value is a numerical outcome of a random phenomenon

ex1 The sum of the pips on two diceex2 The net gain or loss after a game of blackjackex3 The amount of soda in a cup from a vending machine

Discrete Random Variable

• For a discrete random variable X, the number of outcomes is countable (finite)

• Ex3 (the soda from a vending machine) is not discrete. Why?


• Discrete random variables are often displayed using a “probability distribution”

Value of X: x1 x2 x3 … xn

Probability: p1 p2 p3… pn





P(X=x1)=p1

“Probability that X = x1 is p1”





Must add to ‘1’p1 + p2 + … + pn = 1


• A probability distribution for a 6-sided die

• This isn’t a fair die!

X 1 2 3 4 5 6

P(X) 1/3 1/6 1/6 1/6 1/12 1/12

Discrete Random Variables

• Another method of displaying a distribution is with a probability histogram



Each bar is centered on the numerical outcome



The height of the bar is the probability of each outcome



This distribution does not favor any particular outcomes



This distribution ‘favors’ low outcomes


STEPS (1) Define the random variable(2) List all possible event(3) Compute the probability of each

event(4) Display the distribution


• Let look at the probability distribution for the number of ‘heads’ produced by flipping two fair coins


(1) Define the random variable

“Let X = the number of heads produced by flipping two fair coins”


(2) List all possible outcomes• Remember that each coin is

independent• Tip: be systematic when listing

outcomes• Let’s look at the outcomes that have

‘0 heads’TT there’s only one of these


• How many outcomes have ‘1 heads’

HTTHtwo outcomes!


• How many outcomes have ‘2 heads’

HH

only one outcome

• There were 4 equally likely outcomes


(3) Compute the probability of each event

“# outcomes for event/# of outcomes

P(X = 0) = ¼P(X = 1) = 2/4 = ½ P(X = 2) = ¼


(4) Display the distribution

Probability Distribution

x 0 1 2

P(X = x) ¼ ½ ¼


(4) Display the distribution

Probability Histogram

.6

.4

.2

0 1 2 3

Continuous Random Variables

• Unlike the discrete random variable, a continuous random variable has an uncountable number of outcomes

• A continuous random variable X takes on all values in an interval of numbers


• The probability distribution for a continuous rand var is described/ displayed with a density curve.

• The probability of an event is the area under the density curve and above the values of X that make up the event

• The area under a density curve is ‘1’• A density curve lies above the x-axis at

all points.


These distributions are called “uniform”



• Curiously, since there is no area above any single value of x, P(X = x) = 0 for all values of x


• The Normal distribution is also a density curve!

• P( - < z < ) = 1• If a random variable can be

transformed into the Normal distribution, we can use methods we learned earlier to compute probability!


EXAMPLEA soda machine dispenses soda into a 12 oz paper cup. The amount of soda dispensed if Normal with = 11.8 oz and = 0.14 oz. What proportion of cups dispensed are between 11.7 and 12 ozs?


Remember these:1. State problem in terms of ‘x,’ the observed

variable. Draw and shade the distribution of ‘x.’

2. Standardize (find the z-score). Use proper notation!Draw and shade the Normal distribution.

3. Find the area using Table A or your calculator.Remember that the area under curve is ‘1.’

4. Write up a conclusion in context of the problem.


(1) This time, we define our random variable X.“Let X = the amount of soda dispensed into a cup.”

11.66 11.8 11.94 11.7 12


(2) Standardize (find the z-score). Draw and shade the Normal distribution.

11.7 11.8 12 11.811.7 12

0.14 0.14

0.71 1.43

P X P z

P z

-0.71 1.43


(3) Find the area using Table A or your calculator.Remember that the area under curve is ‘1.’

P(-0.71<z<1.43) = 0.6848

(4) Write up a conclusion in context of the problem.

“The proportion of cups whose volume of soda is between 11.7 and 12 ozs is 0.6848”

7.2 MEANS AND VARIANCES OF RANDOM VARIABLES

Mean of a Random Variable

• Symbol is ‘mu’ (again)• This mean is often called the

“expected value” Ex

• The expected value does not have to be a possible value of X because it describes behavior over the long run


• Discrete Random Variables:The mean is the sum of the products of the values and their probability

• X = x1p1 + x2p2 + x3p3 + … + xnpn

• X = xipi


Continuous Random Variables:• We use techniques from chapter 2 to

describe the relative location of the mean (depending on the shape of the density curve)

• Unless the curve is “well known” (i.e.Normal), we will not find the actual numerical value of the mean.

Variance of a Random Variable

Discrete Random Variables• X

2 = (x1 - X)2·p1 + (x2 - X)2·p2 + (x3 - X)2·p3 + … + (xn - X)2·pn

• X2 = (xi - X)2·pi

• Yes, this formula is quite long and tiresome

• Calculator talk:If L1 = xi and L2 = pi, then X

2 = sum(L1 - X)2·L2

Statistical Estimation

• Here’s the problem:We would like to know the mean height of American males. BUT, we cannot have a census.

• Solution:take a sample of American males and compute the mean height of the sample.

Statistical Estimation

• Suppose we took many samples.• The histogram for the means of the

samples will have it’s own distribution• remember, this is the distribution of

the means of the samples, and not the distribution of the heights of American males

• This distribution of the samples is called a “sampling distribution”

The Law of Large Numbers

• For any distribution, as the number of observations increases in the sample, the sample mean (Xbar) will approach the population mean .

• This is true for all distributions, not just Normal distributions.

The Law of Large Numbers

Remember the following for random behavior:

1. Behavior in the short run is unpredictable2. Behavior in the long run is regular and

predictableThis is the Law of Large Numbers

This begs the question, how many observations is “the long run?”

Transformations for Means

• If a random variable undergoes a linear transformation to form a new random variable,

• Then the mean of the new random variable undergoes the same transformation.

• If: X2 = a + bX1

• Then: X2 = a + b X1

Combining Two Means

• If two random variables are added/subtracted to for a new random variable,

• Then the mean of the new random variable is the two means added/subtracted

• IF: X = X1 ± X2

• Then X = X1 ± X2

Transformations for Variance

• If a random variable undergoes a linear transformation,

• Then the variance of the new random variable is the product of the square of the multiplier and the old variance

• If: X2 = a + bX1

• Then: (X2)2 = b2· (X1)2

• Std Dev is the square root of variance

Combining Two Variances

• If two random variables are added/subtracted to for a new random variable,

• Then the variance of the new random variable is the sum of the squares of the old variances

• IF: X = X1 ± X2

• Then (X)2 = (X1)2+ (X2)2

• ALWAYS ADD VARIANCES• Std Dev is the square root of variance

Normal Random Variables

• Any linear combination or transformation of Normal random variables is also Normal.

• IMPORTANT: you must state that a distribution is Normal before you can use techniques for Normal density curves.– Especially true after a linear

combination or transformation.

WILSON!!!!I t ’ s d o u b t f u l t h a t t h i s i s N o r m a l b e h a v i o r

Stats chapter 7

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