9COMPUTE probabilities using the probability distribution of a discrete random variable. 9CALCULATE and INTERPRET the mean (expected value) of a discrete random variable. 9CALCULATE and INTERPRET the standard deviation of a discrete random variable. 9COMPUTE probabilities using the probability distribution of certain continuous random variables.

AP Stats ~ Lesson 6A: Discrete and Continuous Random Variables

Objectives:

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

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

X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X = 3: HHH

Value 0 1 2 3 Probability 1/8 3/8 3/8 1/8

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.

There are two main types of random variables: discrete and continuous. Discrete random variables are variables where we can list all possible outcomes for a random variable and assign probabilities to each one.

Example: Imagine selecting a U.S. high school student at random. Define the random variable X = number of languages spoken by the randomly selected student. The table below shows us the probability distribution of X, based on a sample of students from the U.S. Census at School database. Languages: 1 2 3 4 5 Probability: 0.630 0.295 0.065 0.008 0.002 a) Show that the probability distribution for X is legitimate.

b) Make a histogram of the probability distribution and describe what you see.

c) What is the probability that a randomly selected student speaks at least 3 languages?

Example: North Carolina State University posts the grade distributions for its courses online. Students in Statistics 101 in a recent semester received 26% A’s, 42% B’s, 20% C’s, 10% D’s, and 2% F’s. Choose a Statistics 101 student at random. The student’s grade on a four-point scale (with F=0 and A=4) is a discrete random variable X. a) Make a probability distribution for X.

b) Say in words what the P(X is greater than 3) means. What is this probability?

c) Say in words what the P(X is greater than or equal to 3) means. What is this probability?

d) Write the event “the student got a grade worse than a C” in terms of values of the random variable X. What is this probability?

Mean (Expected Value) of 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.

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: ¦

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Example: Suppose that a person is playing roulette. There are 38 slots on the roulette wheel - 18 roulette slots are red, 18 are black, and 2 are green. Suppose the person places a $1 bet on red. If the ball lands in a red slot, the person will get his original dollar back, plus another dollar. If it lands on black or green, the person will lose his dollar. Define X= net gain from a single $1 bet. a) What is the probability distribution for X?

b) What is the player’s average gain (expected value)? (Remember, this is not the expected gain on a single bet, but it is the expected gain over many bets.)

Example: In a previous example, we talked about the number of languages a random high school student would speak, and we had the following probability distribution: Languages: 1 2 3 4 5 Probability: 0.630 0.295 0.065 0.008 0.002 Compute the mean of the random variable X, and interpret the value in the context of the problem.

Standard Deviation of a Discrete Random Variable

Since we use the mean as the measure of center for a discrete random variable, we use the standard deviation as our measure of spread. The definition of the variance of a random variable is similar to the definition of the variance for a set of quantitative data.

Suppose that X is a discrete random variable whose probability distribution is Value: x1 x2 x3 … Probability: p1 p2 p3 … and that µX is the mean of X. The variance of X is

Var(X) = X2 = (x1 X )2 p1 + (x2 X )2 p2 + (x3 X )2 p3 + ...

= (xi X )2 pi

To get the standard deviation of a random variable, take the square root of the variance.

Example: Let’s return again to our foreign-language speaking students. We just calculated the mean to be 1.457. Compute and interpret the standard deviation of the random variable X. Languages: 1 2 3 4 5 Probability: 0.630 0.295 0.065 0.008 0.002

NOTE: there are instructions for using your calculator to find the mean and standard deviation on page 354 in your book. However, you must still show how you set up the problem if you use the calculator. Calculator calls alone are not enough!

Discrete random variables commonly arise from situations that involve counting something. Situations that involve measuring something often result in a continuous random variable.

The probability model of a discrete random variable X assigns a probability between 0 and 1 to each possible value of X. A continuous random variable Y has infinitely many possible values. All continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability.

Look at the following examples. In each case, decide which is discrete and which is continuous:

•Shoe size vs. foot length

•The amount of time it takes to run the 110 meter hurdles vs. the number of hurdles jumped.

•A student’s age vs. the number of birthdays they’ve had

Example: Suppose we want to choose a number between 0 and 1, allowing ANY number between 0 and 1 to be a valid outcome. We consider all numbers to be equally likely. A) Draw a density curve that models the probability distribution.

B) Find the following probabilities: P(0.2 < X < 0.7) P( X > 0.65) P( X = 0.126)

Example: Normal probability distributions. The heights of young women closely follow the Normal distribution with mean µ = 64 inches and standard deviation σ = 2.7 inches. Now choose one young woman at random. Call her height Y. If we repeat the random choice very many times, the distribution of values of Y is the same Normal distribution that describes the heights of all young women.

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Problem: What’s the probability that the chosen woman is between 68 and 70 inches tall?

Step 1: State the distribution and the values of interest. The height Y of a randomly chosen young woman has the N(64, 2.7) distribution. We want to find P(68 ≤ Y ≤ 70).

Step 2: Perform calculations—show your work! The standardized scores for the two boundary values are:

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Example: The weights of 3-year-old females closely follow a Normal distribution with a mean of 30.7 pounds and a standard deviation of 3.6 pounds. Randomly choose one 3-year-old female and call her weight X. What is the probability that a randomly selected 3-year-old female weighs at least 30 pounds?

HOMEWORK:

Page 329: #1-25 odds, 27-30 Read pp. 363-382