4.1 Probability Distributions

1. 4.1 Probability Distributions Discrete and continuous random variables Constructing discrete probability distributions and their graphs Verifying probability distributions Mean, variance, and standard deviations of discrete probability distributions Expected values

2. Random variable A random variablex represents a numerical value associated with each outcome of a probability experiment.

3. Discrete A random variable is discrete if it has a finite or countable number of possible outcomes that can be listed.

4. Continuous A random variable is continuous if it has an uncountable number of possible outcomes, represented by an interval on the number line.

5. Try it yourself 1 • Discrete and Continuous Variables Decide whether the random variable x is discrete or continuous. Explain your reasoning. • Let x represent the speed of a Space Shuttle. • Let x represent the number of calves born on a farm in one year. Continuous Discrete

6. Discrete Probability Distribution A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A discrete probability must satisfy the following conditions.

7. Try it yourself 2 • Constructing and Graphing a Discrete 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. Construct and graph a probability distribution.

8. Try it yourself 2

9. Try it yourself 3 • Verifying Probability Distributions Verify that the distribution you constructed in Try It Yourself 2 is a probability distribution. Each P(x) is between 0 and 1 and ∑P(x)=1. Because both conditions are met, the distribution is a probability distribution.

10. Try it yourself 4 • Identifying Probability Distributions Decide whether the distribution is a probability distribution. Explain your reasoning. Each P(x) is between 0 and 1 and ∑P(x)=1. Because both conditions are met, the distribution is a probability distribution.

11. Try it yourself 4 • Identifying Probability Distributions Decide whether the distribution is a probability distribution. Explain your reasoning. Each P(x) is between 0 and 1 and ∑P(x)=1. Because both conditions are met, the distribution is a probability distribution.

12. Mean The mean of a discrete random variable is given by μ = ∑ x P(x). Each value of x is multiplied by its corresponding probability and the products are added.

13. Try it yourself 5 • Finding the Mean of a Probability Distribution Find the mean of the probability distribution you constructed in Try It Yourself 2. What can you conclude?

14. Try it yourself 5 μ = 2.6 On average, a new employee makes 2.6 sales per day.

15. Variance The variance of a discrete random variable is σ² = ∑(x-μ)² P(x).

16. Standard deviation The standard deviation is σ = √σ² = √ ∑(x-μ)² P(x).

17. Try it yourself 6 • Finding the Variance and Standard Deviation Find the variance and standard deviation of the probability distribution constructed in Try It Yourself 2.

18. Try it yourself 6

19. Expected Value The expected value of a discrete random variable is equal to the mean of the random variable. Expected Value = E(x) = μ = ∑x P(x)

20. Try it yourself 7 • Finding an Expected Value At a raffle, 2000 tickets are sold at $5 each for five prizes of $2000, $1000, $500, $250, and $100. You buy one ticket. What is the expected value of your gain? μ = -$3.08 Because the expected value is negative, you can expect to loss an average of $3.08 for each ticket you buy.