121616949-math.227 - In particular if Z I f x dx diverges...

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9.8 Probability 213 that the cumulative distribution function , F ( x ) = P ( X x ) = Z x -∞ f ( t ) dt, is continuous. Hence, we say that X is a continuous random variable. Note that because the area does not decrease as x increases, the function F ( x ) is non-decreasing. The entire collection of probabilities for a random variable X , namely P ( X x ) for all x , is called a probability distribution . EXAMPLE 9.24 Suppose that a < b and f ( x ) = ( 1 b - a if a x b 0 otherwise. Then f ( x ) is the uniform probability density function on [ a, b ]. and the corresponding distribution is the uniform distribution on [ a, b ]. The next theorem is not hard to prove; if I is an interval [ a, b ], we write Z I f ( x ) dx for Z b a f ( x ) dx . THEOREM 9.25 Let I be an interval. If 0 f ( x ) g ( x ) for x I and f and g are integrable on I then 0 Z I f ( x ) dx Z
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Unformatted text preview: . In particular, if Z I f ( x ) dx diverges, then so does Z I g ( x ) dx while if Z I g ( x ) dx converges, then so does Z I f ( x ) dx . Note that the theorem applies to infinite intervals, in which case the integrals are improper integrals. EXAMPLE 9.26 Notice that 0 ≤ f ( x ) = e-x 2 / 2 ≤ e-x/ 2 for | x | > 1. It is easy to check that Z ∞ 1 e-x/ 2 dx < ∞ , so by the theorem Z ∞ 1 e-x 2 / 2 dx is finite. Since f is symmetric around the y-axis, Z-1-∞ e-x 2 / 2 dx is also finite, and so finally A = Z ∞-∞ e-x 2 / 2 dx = Z-1-∞ e-x 2 / 2 dx + Z 1-1 e-x 2 / 2 dx + Z ∞ 1 e-x 2 / 2 dx...
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