Chapter 3 Discrete Random Variables and Probability Distributions

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Presentation transcript:

Chapter 3 Discrete Random Variables and Probability Distributions 3.2 - Probability Distributions for Discrete Random Variables 3.3 - Expected Values 3.4 - The Binomial Probability Distribution 3.5 - Hypergeometric and Negative Binomial Distributions 3.6 - The Poisson Probability Distribution

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Suppose X is transformed to another random variable, say h(X). Then by def, Examples: “Moment-generating function” “Characteristic function” (Discrete Fourier Transform)

b General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Suppose X is constant, say b, throughout entire population… b Then by def,

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Suppose X is constant, say b, throughout entire population… Then…

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… Then… i.e.,…

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… Add any constant b to X… Then… i.e.,…

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… Add any constant b to X… Then… i.e.,…

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… then X is also multiplied by a.

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… then X is also multiplied by a. i.e.,… i.e.,…

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Add any constant b to X… then b is also added to X .

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X Add any constant b to X… then b is also added to X . i.e.,… i.e.,…

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X

General Properties of “Expectation” of X POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 random variable X Discrete General Properties of “Expectation” of X This is the analogue of the “alternate computational formula” for the sample variance s2.