Question

1. (10 points) Let f(x, y, z, t) = e-z-y-z-t, x > 0, y > 0, 2 > 0,t > 0, and =0 otherwise, be the joint PDF of (X, Y, Z,T) (a) Compute P{X<Y <T<2} (b) Compute P {X = T = 2 = Y} (c) Compute E[X + 2Y + 32 +T]

(d) Are X,T,Y,Z are mutually independent? Explain why they are indepedent or why they are not independent.

(e) Find the pdf of K, where K=X+T+Y

Answer 1

Complete solution for the problem can be found in the images below with all the necessary steps.
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- У-у-2-t (717А? 1). 1) 5 e d td z dy dx. + Во -x-y-t é at dzaydı 2 9 o6 *-1-t -t e d Edy du О. ok 12 И. alyda e e о ю -4-2 - 14 к e Н dy du. 2 д 59 - e dy gdn e

-X Ia - 3x e 3 da. 2 0 -uu du T 6 x e d + K 6 24 - P(X2Y<Z2T) 그 24 (Ans) 30 P(X=T=Z=Y) Since pointwise probability of Continuous variables is zero. so consider just the event x=1= X-T=0 X-T is continuou random variable of P(x-1) = P(x-1)=0) = Now

E ( x + 2 y + 32+1) = E(x) +2«E(Y) + 3 E(2)+CY as joint pay of -x-y-z-t X, Y, Z, I is ay Now e لاع t -z ке a eae e written A X, Y, Z,I are independent as their joint pay can be as probct of marginal pays. .._X,Y,ZT ~ exponential (1) independently " E (* )=E(Y) = [2)=E(T) = 1 E (x+2 Y+32 +1) = 7.

Yes independent a2 are Yes X, Y, Z, I they all jid exponentiall 1) variable & their joint bay is their marginal pays. product of @ k=X+T+Y X, T,Y tid X+T+ Y ~ Exponential (1) ~ Gamma (3, 1). ~ Gamma (3,1) Sum of ind exponential (0) 'n random variable Gamma (nd) variable. K [ is q