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160a_ho1 - Pstat160a Handout 1 Review of discrete...

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Pstat160a Handout 1 Review of discrete probability distributions I. Probability distributions Recall that the probability distribution function ( p.d.f for short) of a discrete random variable X is p X ( k ) = P ( X = k ) for k in the range of X (typically a subset of 0 , 1 , . . . ) satisfies X k p X ( k ) = 1 . The Cumulative distribution function ( C.d.f. ) F X ( x ) is defined by F X ( x ) = P ( X x ) = X k x P X ( k ) for any x R Properties: - F X is increasing to 1 as x tends to + , starting at its minimum value 0. - F X is a step function - F X is right continuous - If F X has a jump at k , the size of this jump is the probability that X = k When there is no danger of confusion about the random variable, we will often drop the subscript X in the p.d.f and c.d.f. (i.e denote p X ( k ) = p ( k ) , F ( x ) = F X ( x )). - 6 1 x p X ( k ) k x 6 P ( X x ) Remark: The probability distribution and the random variable itself are different albeit related concepts. The p.d.f gives only the probability of different outcomes and the corresponding random variable is implicit, whereas the actual random variable gives a description of the experiment itself. However, in practice and in most of what we will do in this class, only the distribution matters. Because of that, and since p.d.f and C.d.f can be recovered from each others, often one talk about “a random variable X with distribution...”, where either the p.d.f or the c.d.f is specified. Examples: 1) Consider the experiment of rolling a 6-faced dice at random, and let X = number that shows up. Then X has the uniform distribution on 1 , · · · , 6, that is P X ( k ) = 1 n , k = 1 , · · · , n 2) Consider the experiment of selecting 4 cards in a deck of 52 at random, and let X = # of spades. Then X has the following distribution ( Hypergeometric ) P X ( k ) = ( 13 k )( 39 4 - k ) ( 4 52 ) , k = 0 , · · · , 4 In these two examples, note how X refers to a specific experiment, whereas the distribution only give the range of X and the value of the respective probabilities. The process, starting from the description of the experiment, of finding the random variable and its distribution is called (stochastic) modeling . II. Main discrete distributions 1) Hypergeometric: Set-up: Sampling without replacement : There are N objects of two kinds, N 1 and N 2 of each, and n N are selected. Let X = # of the first kind in the selection. P.d.f.: p X ( k ) = ( N 1 k )( N 2 n - k ) ( N n ) 0 k N 1 0 n - k N 2 1
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2 2) Binomial: B ( n , p ) Set-up: There are n independent trials , and for each of those there are only two outcomes that we call S = success or F = failure. The probability of success is P ( S ) = p . Sampling with replacement is an example of Binomial problem. We let X = # of successes in the n trials. P.d.f.: p X ( k ) = n k p k (1 - p ) n - k k = 0 , 1 , . . . , n. 3) Geometric: G ( p ) Set-up: Same as in the Binomial, but the number of trials is not fixed and X = trial # of the first success.
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