Distributions and expected value

1. Distributionsandexpectedvalue Onur DOĞAN

2. RandomVariable Random Variable. Let S be the sample space for an experiment. A real-valued functionthat is defined on S is called a random variable.

3. Distributions • Probability Distributions

4. Discrete Distributions

5. Example 1 • Let 4 coins tossed, and let X be the number of heads that are obtained. Let us find the distributions of that experiment.

6. Bernoulli Distribution Bernoulli Distribution/Random Variable. A random variable Z that takes only twovalues 0 and 1 with Pr(Z = 1) = p has the Bernoulli distribution with parameter p. We also say that Z is a Bernoulli random variable with parameter p.

7. Uniform Distributions on Integers Let a ≤ b be integers. Suppose that the value of arandom variable X is equally likely to be each of the integers a, . . . , b. Then we saythat X has the uniform distribution on the integers a, . . . , b.

8. Uniform Distributions on Integers

9. Binomial Distributions

10. Continuous Distribution Continuous Distribution/Random Variable. We say that a random variable X has acontinuous distribution or that X is a continuous random variable if there exists anonnegative function f , defined on the real line, such that for every interval of realnumbers (bounded or unbounded), the probability thatX takes a value in the intervalis the integral of f over the interval.

11. Continuous Distribution • For each bounded closedinterval [a, b], • Similarly;

12. Continuous Distribution

13. The Expectation of a Random Variable

14. The Expectation of a Random Variable

15. The Expectation of a Random Variable

16. Example

17. Question 1 A shipment of 8 similar microcomputers to contains 3 defective one. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives.

18. Question 2

19. Question 3

21. Question 4