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elemprob-fall2010-page26

elemprob-fall2010-page26 - F y = P X ≤ y In the case of...

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11 Continuous distributions A random variable X is said to have a continuous distribution if there exists a nonnegative function f such that P ( a X b ) = b a f ( x ) dx for every a and b . (More precisely, such an X is said to have an ab- solutely continuous distribution.) f is called the density function for X . Note -∞ f ( x ) dx = P ( -∞ < X < ) = 1. In particular, P ( X = a ) = a a f ( x ) dx = 0 for every a . An example. Suppose we are given f ( x ) = c/x 3 for x 1. Since -∞ f ( x ) dx = 1 and c -∞ f ( x ) dx = c 1 1 x 3 dx = c 2 , we have c = 2. Define F ( y ) = P ( -∞ < X y ) = y -∞ f ( x ) dx . F is called the distri- bution function of X . We can define F for any random variable, not just continuous ones, by setting
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Unformatted text preview: F ( y ) = P ( X ≤ y ). In the case of discrete ran-dom variables, this is not particularly useful, although it does serve to unify discrete and continuous random variables. In the continuous case, the fun-damental theorem of calculus tells us, provided f satisfies some conditions, that f ( y ) = F ( y ) . By analogy with the discrete case, we define the expectation by E X = Z ∞-∞ xf ( x ) dx. In the example above, E X = Z ∞ 1 x 2 x 3 dx = 2 Z ∞ 1 x-2 dx = 2 . 26...
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