Question

X~exp(λ) with λ=1

1) define Y= X^1/2. Find the support of Y and its density.

2) define Z = X^2 + 2X. Find the support of Z and its density.

Answer 1

Given to eap (d)fa=! So f(x)=ě x 220 2 Now Flag a PCX<X) = fitdt et SI-ena F(X) = 1-e? 0 we have to find the density of Y= x². 90 Now let G (Y) Р{ Ч< 9) - P(x ² <9-) = P(x3y31 = Fly2 = l-e-42 from O so g(y)= d Gly) dy dy 11-eyg) PE=0 dey? dy 2 ody dy? -ya Yo g(y) =age

Given z = x2+2x., So G (20 = P(Z <z) ap(*72x sz) EPL x72x+1.32+1) = platu?< 2+1] = P(-2+0 < *+13MZF P[-2+1 -1 5*<V2+ - 1] PE-W2+1 +i) <*30] + +P [Os x</2+1 since xo x2 so x2+2x 2x so x2+2x> o So from Wence PC-12+1 +15KSO] -o. GCZ) = P[0<x</2+1 -11 = F(12+1=1) zo -ė (dzki-) -

G(Z) = (W211 -1) Now 962) = d G (2) dz d -dZ+1 -1) [1-6 1 ) tzt1-1) 이 e dz let t=&zt 1-1 then 912) - d dt -t.dt dz e-t dd2+1-1) 1) d2 li etz21 -0] é NZ+1-1) Co a ſzt1 .(2+1) d (2+1) az 2 (2+1) 9(2) = z>0 2/2+1