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I have X and Y as independent random variables. X belongs U(0,1) and Y belongs U(0,a). I need to find the density function of Z = X+Y.

I have solved before using Jacobian and joint pdf when a = 1, and I understand that I need to look at a >= 1 and a < 1. Not sure how to proceed further with that.

Please help.

Thank you.

1 Answers 1

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$f(x) = \int_{-\infty}^{\infty} U_1(t)U_a(x-t) dt$

suppose $a>1$

if $x<1$

$f(x) = \int_{0}^{x} \frac 1a dt = \frac {x}{a}$

if $1

$f(x) = \int_{x-1}^{x} \frac 1a dt = \frac {1}{a}$

if $a

$f(x) = \int_{x-a}^{1} \frac 1a dt = \frac {1+a-x}{a}$

does it change anything when $a<1?$

$x

$f(x) = \int_{0}^{x} \frac 1a dt = \frac {x}{a}$

$a

$f(x) = \int_{x-a}^{x} \frac 1a dt = 1$

$1

$f(x) = \int_{x-1}^{a} \frac 1a dt = \frac {a+1-x}{a}$

  • 0
    Thank you. Just to clarify, you are considering U1(t) = 1 as you start - correct? – 2017-02-23
  • 0
    $U_1(t) = 1$ when t is in $[0,1]$ and $0$ otherwise. $U_a(t) = \frac 1a$ when $t$ is in $(0,a)$ and $0$ otherwise. – 2017-02-23