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For a nonnegative continuous random variable X, [P P(Xu)du = E(X). Note that, X being nonnegative, E(X) is guaranteed to exist, though it may be + ∞. In this exercise, we are going to explore the above equality. (a) For any b> 0, show that *P(X P(X > u)du = b(1 − F(b)) + ["xf(x)dx, - where f (respectively, F) is the pdf (respectively, cdf) of X. (b) Use the result in (a) to show that E(X)= + ∞ implies JP "P(X > u du = [° P P(X>u)du: = +∞. lim b-+∞J (c) If E(X) < + ∞, show that 1 [** P(X > u)du = E(X). lim b(1 F(b)) = 0 and b→+∞

For a nonnegative continuous random variable X, [P P(Xu)du = E(X). Note that, X being nonnegative, E(X) is guaranteed to exist, though it may be + ∞. In this exercise, we are going to explore the above equality. (a) For any b> 0, show that *P(X P(X > u)du = b(1 − F(b)) + ["xf(x)dx, - where f (respectively, F) is the pdf (respectively, cdf) of X. (b) Use the result in (a) to show that E(X)= + ∞ implies JP "P(X > u du = [° P P(X>u)du: = +∞. lim b-+∞J (c) If E(X) < + ∞, show that 1 [** P(X > u)du = E(X). lim b(1 F(b)) = 0 and b→+∞

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.4: Hyperbolas
Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
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For a nonnegative continuous random variable X, S Note that, X being nonnegative, E(X) is guaranteed to exist, though it may be + ∞. In this exercise, we are going to explore the above equality. (a) For any b > 0, show that S P(X > u)du = b(1 − F(b)) + where f (respectively, F) is the pdf (respectively, cdf) of X. lim P(X > u)du = E(X). (b) Use the result in (a) to show that E(X) = + ∞ implies b "P(X > u du = [" b- (c) If E(X) < +∞, show that f [xf(x)da, lim b(1 – F(b)) = 0 and b→+∞ P(X> u) du = = + ∞. P(X > u)du = E(X).
Transcribed Image Text:For a nonnegative continuous random variable X, S Note that, X being nonnegative, E(X) is guaranteed to exist, though it may be + ∞. In this exercise, we are going to explore the above equality. (a) For any b > 0, show that S P(X > u)du = b(1 − F(b)) + where f (respectively, F) is the pdf (respectively, cdf) of X. lim P(X > u)du = E(X). (b) Use the result in (a) to show that E(X) = + ∞ implies b "P(X > u du = [" b- (c) If E(X) < +∞, show that f [xf(x)da, lim b(1 – F(b)) = 0 and b→+∞ P(X> u) du = = + ∞. P(X > u)du = E(X).
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