4055comFX

4055comFX - = 1 for all x ≥ 0 Observe also that P X ≤-1...

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Math 4055/Sengupta Spring, 2002 Comments concerning P [ X t ] and P [ X < t ] The DISTRIBUTION FUNCTION F X for a random variable X is de±ne by F X ( t ) = P [ X t ] Observe that we have here , not < . The probability of the event [ X < t ] can be obtained by as a limit: P [ X < t ] = lim x t - P [ X < t ] = lim x t - F X ( t ) If the function F X happens to be continuous at t then, and only then, P [ X < t ] is equal to F X ( t ). Example . Consider the distribution function given by F X ( x ) = ± 0 if x < - 1 1 3 + 2 3 ( x + 1) 2 if - 1 x 0 The values of F X ( x ) for x > 0 are not stated. This is because F X (0) = 1 3 + 2 3 = 1 which means P [ X 0] = 1, and so F X ( x
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Unformatted text preview: ) = 1 for all x ≥ 0. Observe also that P [ X ≤ -1] = F X (-1) = 1 3 + 0 = 1 3 while, on the other hand, P [ X < x ] = F X ( x ) = 0 for any x < 0. This implies that P [ X =-1] = 1 3 What is happening here is that the function F X has a discontinuity at-1. Let us calculate P [ | X-1 3 | < 1]: P ² ³ ³ X-1 3 ³ ³ < 1 ´ = P ²-2 3 < X < 4 3 ´ = P ² X < 4 3 ´-P ² X ≤ -2 3 ´ = 1-F X µ-2 3 ¶ = 1-² 1 3 + 2 3 1 3 2 ´ 1...
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