6.1AEXPECTED VALUE OF A

DISCRETE RANDOM VARIABLE

AP Statistics

Random Variables and Probability Distributions

A probability model describes the possible outcomes of a chance process and the likelihood that those outcomes will occur.

A random variable takes numerical values that describe the outcomes of some chance process. The probability distribution of a random variable gives its possible values and their probabilities.

X = 0: TTTX = 1: HTT THT TTHX = 2: HHT HTH THHX = 3: HHH

Value 0 1 2 3

Probability 1/8 3/8 3/8 1/8

Consider tossing a fair coin 3 times.Define X = the number of heads obtained

Discrete Random Variables

There are two main types of random variables: discrete and continuous. If we can find a way to list all possible outcomes for a random variable and assign probabilities to each one, we have a discrete random variable.

When analyzing discrete random variables, we follow the same strategy we used with quantitative data – describe the shape, center, and spread, and identify any outliers.

The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability.

Mean (Expected Value) of a Discrete Random Variable

Suppose that X is a discrete random variable whose probability distribution is

Value: x1

x2 x3 …Probability: p1 p2

p3 …

To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products:

ii

x

px

pxpxpxXE ...)( 332211

In English: To find the expected value of a discrete random variable--take the value and multiply that value by its probability of happening. Do this for each value. Add up the products.

Mean (Expected Value) of a Discrete Random Variable

Value 0 1 2 3

Probability 1/8 3/8 3/8 1/8

Consider tossing a fair coin 3 times. Define X = the number of heads obtained

What is the expected value of X?

( )x i iE X x p

1 3 3 1 12( ) 0 1 2 3 1.5

8 8 8 8 8E X

The expected value for the number of heads obtained when tossing a fair coin three times is 1.5 heads.

Apgar Scores

In 1952, Dr. Virginia Apgar suggested 5 criteria for measuring a baby’s health at birth: skin color, heart rate, muscle tone, breathing and response when stimulated. She developed a 0-1-2 scale to rate a newborn on each of the five scales, which gives a whole-number value from 0 to 10. Apgar scores are still used today to evaluate the health of newborns.

What Apgar scores are typical?Let X = Apgar score of a randomly selected newborn

c. Doctors decided that an Apgar score of 7 or higher indicates a healthy baby. What is the probability that a randomly selected baby is healthy?

b. Make a histogram of the distribution. Describe what you see.

a. Show that the probability distribution is legitimate

Apgar Scores

a. Show that the probability distribution is legitimate.The probabilities are all between 0 and 1 andP(X=0) + P(X=1) + P(X=2) + … + P(X=10) = 1

Apgar Scores

b. Make a histogram of the distribution. Describe what you see.

The graph of Apgar scores is skewed left and unimodal. A randomly selected newborn will most likely have an Apgar score on the high end of the scale, which means that the baby was healthy at birth. The median is 8. Apgar scores vary from 0 to 10, but most newborns have scores between 4 and 10.

Apgar Scores

c. Doctors decided that an Apgar score of 7 or higher indicates a healthy baby. What is the probability that a randomly selected baby is healthy?

Apgar Scores

The probability of choosing a baby that is healthy is P(X≥7). P(X≥7) = P(X=7) + P(X=8) + P(X=9) + P(X=10) = 0.099 + 0.319 + 0.437 + 0.053 = 0.908

We’d have about a 91% chance of randomly choosing a healthy baby.

Mean (Expected Value) of a Discrete Random Variable

Compute the mean of the random variable X and interpret this value in context.

The formula for Expected Value is ( )x i iE X x p

The mean Apgar score of a randomly selected newborn is 8.128. This is the average Apgar score of many, many randomly chosen babies.

Practice Problems

pg. 359-360 #1-13 odd