elemprob-page14 - Hypergeometric . Set P ( X = i ) = m i N-...

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Unformatted text preview: Hypergeometric . Set P ( X = i ) = m i N- m n- i N n . This comes up in sampling without replacement: if there are N balls, of which m are one color and the other N- m are another, and we choose n balls at random without replacement, then X represents the probability of having i balls of the first color. 6. Continuous distributions. A r.v. X is said to have a continuous distribution if there exists a nonnegative function f such that P ( a X b ) = Z b a f ( x ) dx for every a and b . (More precisely, such an X is said to have an absolutely continuous distribution.) f is called the density function for X . Note R - f ( x ) dx = P (- < X < ) = 1. In particular, P ( X = a ) = R a a f ( x ) dx = 0 for every a . Example. Suppose we are given f ( x ) = c/x 3 for x 1. Since R - f ( x ) dx = 1 and c Z - f ( x ) dx = c Z 1 1 x 3 dx = c 2 , we have c = 2. Define F ( y ) = P (- < X y ) = R y- f ( x ) dx . F is called the distribution function of X...
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