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Sec. 4.1 Probability Distributions A ___________ ___________ 𝑥 represents a numerical value associated with each outcome of a probability experiment. A random variable is __________ if it has a finite or countable number of possible outcomes that can be listed. A random variable is __________ if it has an uncountable number of possible outcomes, represented by an interval on the number line. Decide whether or not the random variable is discrete or continuous. The number of sales calls The number of hours spent on sales calls The number of stocks in the Dow Jones Industrial Average The volume of water in a bucket The height of a skyscraper The number of windows in a skyscraper Discrete Probability Distributions A ___________ _______________ ______________ lists each possible value the random variable can assume, together with its probability. A probability distribution must satisfy the following conditions. 1. The probability of each value of the discrete random variable is between 0 and 1, inclusive, or 0 ≤ 𝑃(𝑥) ≤ 1. 2. The sum of all the probabilities is 1 or ∑ 𝑃(𝑥) = 1 Construct a probability distribution in the tables for the following scenarios. An industrial psychologist administered a personality inventory test for passiveaggressive traits to 150 employees. Individuals were given a score from 1 to 5, where 1 was extremely passive and 5 extremely aggressive. A score of 3 indicated neither trait. The results are shown below. Frequency Distribution Score, x 1 2 3 4 5 Frequency, 𝑓 24 33 42 30 21 Probability Distribution A company tracks the number of sales new employees make each day during a 100-day probationary period. The results for one new employee are shown at the left. Sales per day, 𝑥 0 1 2 3 4 5 6 7 Number of days, 𝑓 16 19 15 21 9 10 8 2 Probability Distribution Verify that the following are probability distributions. Days of rain, 𝑥 0 1 2 3 Probability, 𝑃(𝑥) 0.216 0.432 0.288 0.064 𝑥 2 3 4 𝑃(𝑥) −0.1 0.9 0.2 𝑥 5 6 7 8 𝑃(𝑥) 1 16 5 8 1 4 1 32 Mean, Variance, and Standard Deviation The ___________ of a discrete random variable is given by 𝜇 = ∑ 𝑥𝑃(𝑥). Note: The mean of the random variable represents the “theoretical average” of a probability experiment and sometimes is not a possible outcome. The ____________ of a discrete random variable is 𝜎 2 = ∑(𝑥 − 𝜇)2 𝑃(𝑥). The ____________ _____________ is 𝜎 = √𝜎 2 = √∑(𝑥 − 𝜇)2 𝑃(𝑥). Use the tables to find the mean and variance of the probability distributions of the personality inventory from the previous example. 𝑥 1 2 3 4 5 𝑥 1 2 3 4 5 𝑃(𝑥) 0.16 0.22 0.28 0.20 0.14 𝑃(𝑥) 0.16 0.22 0.28 0.20 0.14 Find the mean and variance of the probability distribution involving the number of sales of an employee. Expected Value The _______________ ___________ of a discrete random variable is equal to the mean of the random variable. Expected Value = 𝐸(𝑥) = 𝜇 = ∑ 𝑥𝑃(𝑥) Note: Expected value is another word for mean in a probability distribution, but carries a slightly different interpretation. The expected value is what we would expect to happen over thousands of trials of the same game or probability experiment. Find the expected value of the following problems. At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. What is the expected value of your gain? You are playing a game with a friend that involves rolling a normal six-sided die. You pay $3.50 to play the game. After you roll the die, you earn the same number of dollars of the number rolled.