Distributions and expected value Onur DOĞAN

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.

Distributions Probability Distributions

Discrete Distributions

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.

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.

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.

Uniform Distributions on Integers

Binomial Distributions

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.

Continuous Distribution For each bounded closed interval [a, b], Similarly;

Continuous Distribution

The Expectation of a Random Variable

The Expectation of a Random Variable

The Expectation of a Random Variable

Example

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.

Question 2

Question 3

Question 4