elemprob-fall2010-page30

elemprob-fall2010-page30 - The distribution function of a...

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The distribution function of a standard N (0 , 1) is often denoted Φ( x ), so that Φ( x ) = 1 2 π Z x -∞ e - y 2 / 2 dy. Tables of Φ( x ) are often given only for x > 0. One can use the symmetry of the density function to see that Φ( - x ) = 1 - Φ( x ); this follows from Φ( - x ) = P ( Z ≤ - x ) = Z - x -∞ 1 2 π e - y 2 / 2 dy = Z x 1 2 π e - y 2 / 2 dy = P ( Z x ) = 1 - P ( Z < x ) = 1 - Φ( x ) . An example. Find P (1 X 4) if X is N (2 , 25). Answer. Write X = 2 + 5 Z . So P (1 X 4) = P (1 2 + 5 Z 4) = P ( - 1 5 Z 2) = P ( - 0 . 2 Z . 4) = P ( Z . 4) - P ( Z ≤ - 0 . 2) = Φ(0 . 4) - Φ( - 0 . 2) = . 6554 - [1 - Φ(0 . 2)] = . 6554 - [1 - . 5793] . An example. Find c such that P ( | Z | ≥ c ) = . 05. Answer. By symmetry we want c such that P ( Z
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.

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