1. Distributions and expected value
Onur DOĞAN

2. Random Variable
Random Variable. Let S be the sample space for an experiment. A real-valued function that 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 two values 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 a random variable X is equally likely to be each of the integers a, , b. Then we say that 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 a continuous distribution or that X is a continuous random variable if there exists a nonnegative function f , defined on the real line, such that for every interval of real numbers (bounded or unbounded), the probability that X takes a value in the interval is the integral of f over the interval.

11. Continuous Distribution
For each bounded closed interval [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