Only do D please. (a,b,c) is just for reference. For a nonnegative continuous random variable X,SNote 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 thatS P(X > u)du = b(1 − F(b)) +where f (respectively, F) is the pdf (respectively, cdf) of X.limP(X > u)du = E(X).(b) Use the result in (a) to show that E(X) = + ∞ impliesb"P(X > u du = ["b-(c) If E(X) < +∞, show thatf[xf(x)da,lim b(1 – F(b)) = 0 andb→+∞P(X> u) du == + ∞.P(X > u)du = E(X). (d) We asserted in (c) that E(X) < + ∞ implieslim b(1 F(b)) = 0.b→ +∞We are going to show in this part (which consists of multiple subparts) through acounterexample that the converse does not hold, that is,does not necessarily imply E(X) < +∞.(ii) Letlim b(1 - F(b)) = 0b→ +∞(i) Calculate the indefinite integralg(y) =-fa(1 + Iny)(ylny)²∞+4921n2(1+lny)(ylny)20Use the result in (i) to show that g is a pdf.(iii) Use the result in (i) to calculate the cdf G.(iv) Show that-dy.if y ≥ 2if y < 2.lim b(1 - G(b)) = 0.(v) Let the random variable Y have pdf g. Show that E(Y) = + ∞.

When the probability function is defined on the continuous variable is called the continuous…

(d) We asserted in (c) that E(X) < + ∞ implies lim b(1 F(b)) = 0. b→+∞ We are going to show in this part (which consists of multiple subparts) through a counterexample that the converse does not hold, that is, lim b(1 F(b)) = 0 b4+x does not necessarily imply E(X) < +∞. (ii) Let (i) Calculate the indefinite integral g(y) fa = (1 + Iny) (ylny) ² 21n2(1+lny) (ylny)² 0 Use the result in (i) to show that g is a pdf. (iii) Use the result in (i) to calculate the cdf G. (iv) Show that ∞+49 -dy. lim b(1 G(b)) = 0. 1 if y ≥ 2 if y < 2. (v) Let the random variable Y have pdf g. Show that E(Y) = +∞.

(d) We asserted in (c) that E(X) < + ∞ implies lim b(1 F(b)) = 0. b→+∞ We are going to show in this part (which consists of multiple subparts) through a counterexample that the converse does not hold, that is, lim b(1 F(b)) = 0 b4+x does not necessarily imply E(X) < +∞. (ii) Let (i) Calculate the indefinite integral g(y) fa = (1 + Iny) (ylny) ² 21n2(1+lny) (ylny)² 0 Use the result in (i) to show that g is a pdf. (iii) Use the result in (i) to calculate the cdf G. (iv) Show that ∞+49 -dy. lim b(1 G(b)) = 0. 1 if y ≥ 2 if y < 2. (v) Let the random variable Y have pdf g. Show that E(Y) = +∞.

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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