Section Discrete and Continuous Random Variables AP Statistics

AP Statistics, Section 7.1, Part 12 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 7.1, Part 13 Discrete Probability Distribution Table Value of X:x1x1 x2x2 x3x3 …xnxn Probability: p1p1 p2p2 p3p3 …pnpn

AP Statistics, Section 7.1, Part 14 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 7.1, Part 15 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 7.1, Part 16 Probabilities: X: P(X): 1/16 1/4 3/8 1/4 1/ Histogram

AP Statistics, Section 7.1, Part 17 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 7.1, Part 18 What is the average number of heads?

AP Statistics, Section 7.1, Part 19 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 7.1, Part 110 Definition: 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. All continuous random distributions assign probability 0 to every individual outcome.

AP Statistics, Section 7.1, Part 111 Distribution of Continuous Random Variable

AP Statistics, Section 7.1, Part 112

AP Statistics, Section 7.1, Part 113 Example of a non-uniform probability distribution of a continuous random variable.

AP Statistics, Section 7.1, Part 114 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. What is the probability that the wait is between 12 and 20 minutes?

AP Statistics, Section 7.1, Part 115 Density Curve.

AP Statistics, Section 7.1, Part 116 Probability shaded. P(12≤ X ≤ 20) = 0.5 · 8 =.40

AP Statistics, Section 7.1, Part 117 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 7.1, Part 118 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 7.1, Part 119 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 7.1, Part 120

AP Statistics, Section 7.1, Part 121 Example continued Calculations About 21% of sample results will be off by more than two percentage points.

AP Statistics, Section 7.1, Part 122 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 7.1, Part 123 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.