Random Variable A random variable is a function
Random Variable A random variable is a function that assigns a numerical value to each outcome of an experiment. • Consider the experiment of tossing a balanced coin o o Sample space = H (heads), T (tails) Let X=0 if the coin lands on heads, and let X=1 if the coin lands on tails These outcomes each happen with a certain probability P(X=0) = 0. 5 and P(X=1) = 0. 5 • A random variable that has a finite number of possible outcomes is a discrete random variable. 1/10/2022 1
Probability Distribution A discrete probability distribution lists the set of all possible outcomes of a random variable, X, along with the probability that each outcome occurs. • Let X be the random variable associated with flipping a coin. • Probability distribution of X: X P(X) 0 0. 5 1/10/2022 2
Example (5. 3, p. 144) Consider a deck of 52 cards with 13 hearts, 13 spades, 13 clubs, and 13 diamonds and assume that a card is drawn at random. Define a random variable X to be 1 if a heart is drawn, 2 if a spade is drawn, and 3 if either a club or a diamond is drawn. • • List the values of X and their probabilities What is the probability that X is greater than 1? What is the probability that X is either 1 or 3? Provide a graphical view of the probability distribution of X (histogram) 1/10/2022 3
Probability Mass Function (PMF) • 1/10/2022 4
Mean and Variance of a Discrete Random Variable • 1/10/2022 5
Example Prescott National Bank has six tellers available to server customers. The number of tellers busy with customers at, say, 1: 00 pm varies from day to day and depends on chance; so, it is a random variable which we will call X. Past records indicate that the probability distribution of X is as shown (on the next slide). The table indicates, for instance, that the probability is 0. 262 that exactly five tellers will be busy with customers at 1: 00 pm. Find the mean of the random variable X. In other words, on average, how many tellers will be busy at 1: 00 pm? 1/10/2022 6
Example (cont) x P(X=x) 0 0. 029 1 0. 049 2 0. 078 3 0. 155 4 0. 212 5 0. 262 6 0. 215 1/10/2022 7
Mean and Variance of a Discrete Random Variable • 1/10/2022 8
Example (cont) x (x-μ)2 P(X=x) (x-μ)2*P(X=x) 0 -4. 118 16. 958 0. 029 0. 492 1 -3. 118 9. 722 0. 049 0. 476 2 -2. 118 4. 486 0. 078 0. 350 3 -1. 118 1. 250 0. 155 0. 194 4 -0. 118 0. 014 0. 212 0. 003 5 0. 882 0. 778 0. 262 0. 204 6 1. 882 3. 542 0. 215 0. 762 1/10/2022 9
Binomial Distribution Many problems in probability concern the repetition of an experiment having two possible outcomes. Each repetition of the experiment is called a trial. Suppose: • Your experiment consists of n trials. • Each trial has exactly two possible outcomes referred to as a success and a failure. • The n trials are independent. • The probability of a success for each trial is denoted by p and remains the same for each trial. 1/10/2022 10
Binomial Distribution Let the random variable X be the number of successes out of n trials. We say that X has a binomial distribution. 1/10/2022 11
Binomial Distribution Examples: • Testing the effectiveness of a drug: several patients take the drug (the trials), and for each patient, the drug is either effective (success) or not effective (failure). • Weekly sales of a car salesperson: The salesperson has several customers during the week (the trials), and for each customer, the salesperson either makes a sale (success) or does not make a sales (failure). • Taste tests for colas: A number of people taste two different colas (the trials), and for each person, the preference is either for the first cola (success) or for the second cola (failure). 1/10/2022 12
Binomial Coefficient • 1/10/2022 13
Binomial Coefficient Suppose that you flip a coin 10 times. How many ways can you get 2 heads out of the 10 flips? HHTTTT HTHTTTTTTT HTTHTTTTTT HTTTTT …… Etc… 1/10/2022 14
Binomial Coefficient • 1/10/2022 15
Binomial Distribution • 1/10/2022 16
Example According to tables provided by the US National Center for health Statistics in Vital Statistics of the United States, there is about a 60% chance that a person age 20 will still be alive at age 65. Suppose that 3 people age 20 are selected at random. Find the probability that the number alive at age 65 will be a) Exactly two, b) at most one, c) at least one d) Determine the probability distribution of the number alive at age 65 1/10/2022 17
Example • Identify a success o In this problem, a success is that a person currently age 20 will be alive at age 65 • Determine p, the success probability o This is the probability that a person currently age 20 will be alive at age 65, which is 60%. So, p=0. 6 • Determine n, the number of trials o In this case, the number of trials is the number of people in the study, which is three. So, n=3. • What is the binomial probability formula for this problem? P(X=k) = 3 Cx*(0. 6)k * (1 -0. 6)3 -k 1/10/2022 18
Mean and Variance of Binomial Distribution • 1/10/2022 19
Example For the previous example, what is the mean and standard deviation of the random variable? 1/10/2022 20
Example (Using the Binomial Table) Based on data from the Statistical Abstract of the United States, the probability that a newborn baby will be a girls is about 0. 487. Suppose that 10 babies are selected at random. What is the probability that at least 1 will be a girl? What is the probability that exactly half of the babies will be girls? 1/10/2022 21
- Slides: 21