# lec9 - Special discrete pmfs Intuitive idea: In many...

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Special discrete pmfs Intuitive idea: In many theoretical and practical problems, several probability mass functions occur often enough to be worth exploring. Common feature: The sample space is always Fnite or countably many. Example: 1. Bernoulli distribution 2. Binomial distribution 3. Geometric distribution 4. Poisson distribution 5. Compounded distribution 1

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Bernoulli distribution Situation: Bernoulli experiment (only two outcomes: success/failure ) with P ( success ) = p We defne a random variable X as: X ( success ) = 1 X ( Failure ) = 0 The probability mass Function p X oF X is then: p X (0) = 1 p p X (1) = p This probability mass Function is called the Bernoulli mass function . 2
Bernoulli experiment: Examples Toss a coin: Ω = { H,T } Throw a fair die: Ω = { face value is a six , face value is not a six } Sent a message through a network and record whether or not it is received: Ω = { successful transmission , unsuccessful transmission } Draw a part from an assembly line and record whether or not it is defective: Ω = { defective , good } Response to the question “Are you in favor of the above measure”? (in reference to a new tax levy to be imposed on city residents), in a randomized survey conducted via telephone: Ω = { yes , no } 3

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Bernoulli random variable (cont’d) The cdf of the Bernoulli distribution , F X is then: F X ( t ) = P ( X t ) = 0 t < 0 1 p 0 t < 1 1 1 t This function is called the Bernoulli cumulative distribution function (cdf) . Expected value and Variance of a Bernoulli random variable: E ( X ) = x p X ( x ) = 0(1 p ) + 1 · p = p Var ( X ) = x ( x E ( X )) 2 p X ( x ) = (0 p ) 2 · (1 p ) + (1 p ) 2 · p = p (1 p ) 4
Sequence of Bernoulli Experiments

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## This note was uploaded on 02/01/2012 for the course STAT 330B taught by Professor Zhou during the Spring '11 term at Iowa State.

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lec9 - Special discrete pmfs Intuitive idea: In many...

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