AP Statistics, Section 7.11 The Practice of Statistics Third Edition Chapter 7: Random Variables 7.1 Discete and Continuous Random Variables Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

AP Statistics, Section 7.12 Essential Questions What is discrete random variable? What is a probability distribution? How do you construct a probability distribution for a discrete random variable? Given a probability distribution for a random variable, how do you construct a probability histogram? What is a density curve? What is a uniform distribution? What is a continuous random variable and how do you define a probability distribution for a continuous random variable?

AP Statistics, Section 7.13 Random Variables A random variable is a variable whose value is a numerical outcome of a random phenomenon. For example: Flip three coins and let X represent the number of heads. X is a random variable. We usually use capital letters to denotes random variables. The sample space S lists the possible values of the random variable X. We can use a table to show the probability distribution of a discrete random variable.

AP Statistics, Section 7.14 Discrete Probability Distribution Table Value of X:x1x1 x2x2 x3x3 …xnxn Probability: p1p1 p2p2 p3p3 …pnpn

AP Statistics, Section 7.15 Discrete Random Variables A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities. X: x 1 x 2 x 3 … x k P(X): p 1 p 2 p 3 … p k 1. 0 ≤ p i ≤ 1 2. p 1 + p 2 + p 3 +… + p k = 1.

AP Statistics, Section 7.16 Probability Distribution Table: Number of Heads Flipping 4 Coins TTTT TTTH TTHT THTT HTTT TTHH THTH HTTH HTHT THHT HHTT THHH HTHH HHTH HHHT HHHH X01234 P(X)1/164/166/164/161/16

AP Statistics, Section 7.17 Probabilities: X: P(X): 1/16 1/4 3/8 1/4 1/ Histogram

AP Statistics, Section 7.18 Questions. Using the previous probability distribution for the discrete random variable X that counts for the number of heads in four tosses of a coin. What are the probabilities for the following? P(X = 2) P(X ≥ 2) P(X ≥ 1) = =.9375

AP Statistics, Section 7.19 What is the average number of heads?

AP Statistics, Section Continuous Random Varibles Suppose we were to randomly generate a decimal number between 0 and 1. There are infinitely many possible outcomes so we clearly do not have a discrete random variable. How could we make a probability distribution? We will use a density curve, and the probability that an event occurs will be in terms of area.

AP Statistics, Section Distribution of Continuous Random Variable

AP Statistics, Section Problem Let X be the amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train. Suppose that the density curve is a uniform distribution. Draw the density curve for 0 to 20 minutes. What is the probability that the wait is between 12 and 20 minutes?

AP Statistics, Section Density Curve.

AP Statistics, Section Probability shaded. P(12≤ X ≤ 20) = 0.5 · 8 =.40

AP Statistics, Section Normal Curves We’ve studied a density curve for a continuous random variable before with the normal distribution. Recall: N(μ, σ) is the normal curve with mean μ and standard deviation σ. If X is a random variable with distribution N(μ, σ), then is N(0, 1)

AP Statistics, Section Example Students are reluctant to report cheating by other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?” Suppose that if we could ask all undergraduates, 12% would answer “Yes.” The proportion p = 0.12 would be a parameter for the population of all undergraduates.

AP Statistics, Section Example continued Students are reluctant to report cheating by other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?” What is the probability that the survey results differs from the truth about the population by more than 2 percentage points? Because p = 0.12, the survey misses by more than 2 percentage points if

AP Statistics, Section 7.118

AP Statistics, Section Example continued Calculations About 21% of sample results will be off by more than two percentage points.

AP Statistics, Section Summary A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities. A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The Probability of any event is the area under the density curve and above the values of X that make up the event.

AP Statistics, Section Summary When you work problems, first identify the variable of interest. X = number of _____ for discrete random variables. X = amount of _____ for continuous random variables.