Discrete Probability Distributions Section 5.2

Objectives Distinguish between discrete and continuous random variables. Graph discrete probability distributions. Compute  and  for a discrete probability distribution. Compute  and  for a linear function of a random variable x.

Random Variables

Random Variables A discrete random variable will be the result of a count. Example The number of students in a statistics class is a discrete random variable. 15, 25, 50, and 250 are all possible. However, 25.5 students is not possible for the number of students.

Random Variables The continuous random variables will be the result of a measurement on a continuous scale. Example Air pressure in an automobile tire is a continuous random variable. The air pressure could, be any value from 0 psi to the bursting pressure of the tire. Values such as 20.126 psi, 20.12678 psi, and anything else is possible.

Probability Distribution of a Discrete Random Variable A probability distribution is an assignment of probabilities to each distinct value of a discrete random variable or to each interval of values of a continuous random variable. Features of the probability distribution of a discrete random variable… Has a probability assigned to each distinct value of the random variable. 2) The sum of all assigned probabilities must be 1.

Example–Discrete probability distribution A test was developed to measure boredom tolerance. It was administered it to a group of 20,000 adults. The possible scores were 0, 1, 2, 3, 4, 5, 6; with 6 indicating the highest tolerance for boredom. Find P(x) for each score. Then make a relative frequency histogram

Probability Distribution of Test Scores Example–Discrete probability distribution Probability Distribution Probability Distribution of Test Scores

Example–Discrete probability distribution The Topnotch Clothing Company needs to hire someone with a score on the boredom tolerance test of 5 or 6 to operate the fabric press machine. The probability that someone in the group who took the boredom tolerance test made either a 5 or 6 is … 1 out of 10 of the group who took the boredom tolerance test would qualify at Topnotch Clothing.

Probability Distribution of a Discrete Random Variable A probability distribution can be thought of as a relative-frequency distribution. As such, it has a mean and standard deviation. Greek letters used, information given is from the entire population.

Standard deviation of x… Probability Distribution of a Discrete Random Variable Expected value of x… Standard deviation of x… x is the value of a random variable P(x) is the probability of the variable Summation is each of the values of the variable

Probability Distribution of a Discrete Random Variable The mean of a probability distribution is often called the expected value of the distribution. The standard deviation is often represented as a measure of risk. A larger standard deviation implies a greater likelihood that the random variable x is different from the expected value .

Example–Expected value, standard deviation Are we influenced to buy a product by an ad we saw on TV? Compute the mean and standard deviation of the distribution.

Example–Solution x (#of viewings) 𝑃(𝑥) 𝑥𝑃(𝑥) 𝑥−𝝁 (𝑥−𝝁) 𝟐 (𝑥−𝝁) 𝟐 𝐏(𝐱) 1 .27 .27 -1.54 2.372 0.640 .31 2 .62 -0.54 0.292 0.091 3 .18 .54 0.46 0.212 0.038 4 .09 .36 1.46 2.132 0.192 5 .15 .75 2.46 6.052 0.908

Example–Solution The average number of times a buyer views the infomercial before purchase is sum of column 3. To find the standard deviation, take the square root of the sum of column 6.

5.1 Introduction to Random Variables and Probability Distributions Summarize Notes Read section 5.1 Homework Pg.190 #1, 5, 6, 8-10, 13, 15, 16