For a nonnegative continuous random variable X, 5. 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 > u)du = b(1 − F(b)) + [*xƒ(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 cb 8 "P(X > u du = [1 P(X> u)du = +∞. lim 6+∞ √0 (c) If E(X) < + ∞, show that SOF lim b(1 F(b)) = 0 and b→+∞ P(X> u)du = E(X).
For a nonnegative continuous random variable X, 5. 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 > u)du = b(1 − F(b)) + [*xƒ(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 cb 8 "P(X > u du = [1 P(X> u)du = +∞. lim 6+∞ √0 (c) If E(X) < + ∞, show that SOF lim b(1 F(b)) = 0 and b→+∞ P(X> u)du = E(X).
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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