Question

Let X and Y ~U(0, 1]. X and Y are independent

a) Find the PDF of X+Y

b) Suppose now X~(0, a] Y~(0,b] and $0<a\leq b$. Find the PDF of X+Y

Answer 1

i have done if . firstly i answered the question 1 and then question 2, please go through the images for the solutions of the problems .

ut, x and y muco,t) .X, Y independent, a. 2 = x+y, let, Now, fx(x) = fy (Y) = { 1 if ocard loo.w. rify(@)= a, acxcb.) when, an ucab). z = x ty ... X = f(z -y Now, the fz (2) is, f₂ (2) = fx(z-y): fy (Y) dy. Now, Z=xty ... x= Now, as fon O2 y 22, fx (4) at and 0 0.W. so here, fa (2) = f (2-by) dy never Now, the integral is o, su unless OLZ-y L L. 7 Z-LYLZ and then it ift.

Now, max name of 2Exty will be a [when, x=1, 4=t]. and minimum value will be 22.05 when x=0, y =o) Now, for OcZ 22, we will calculate the .p. dit Now, when, his OLZ LL, non fa(z) = J ay = Z CBN when & LZ22 fz (2) = 1 dlh 2-z. STO

VCOV & (2) = { z if olaLL a-z if LOLZ L2 to other wise 2) if anuco, as you v co b] - then, fx(x) = f 40 La La o o.W. and, fy (y) = 1 if oly Lb oo.co. NON Now, z=xty. the maximum value will be q=ath when x=ay=b and minimum value will be 2=0 when, x=0, 420.

for, olz-yla. z-aLYLZ. Now, for, olzla f₂ (2) = f f (2-4) dy for alzab, f (2) = 1 fez-y) dy max Ca, b-a) for, blzzatb, 22 (2) = af (2-4) dy J b-a = 2b a .

. The pdf of zis. f(2)=( f (z-g) ay if ocala f f(2-y) ay if azzub maxca b-a) .. 2b it blzL atb O O.W.