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 &lt; x ] = F X ( x ) = 0 for any x &lt; 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 | &lt; 1]: P X-1 3 &lt; 1 = P -2 3 &lt; X &lt; 4 3 = P X &lt; 4 3 -P X -2 3 = 1-F X -2 3 = 1- 1 3 + 2 3 1 3 2 1...
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