Vtyjkyuk

I have problems with deriving the CDF of a weighted difference of iid uniformly distributed random variables.

Assume that $X_1$ and $X_2$ $stackreliidsim U[0,1]$. Define Z = $acdot X_1 - X_2$, where $a > 0$. How can I derive $P(Z<z)$? I was struggling with applying the general definition of a convolution on my particular case...

Thanks in advance!

On the one hand we have $$P(A) = E[1_A]$$ on the other hand we have $$E[g(X_1,X_2)] = int g(x_1,x_2) f_(X_1,X_2)(x_1,x_2) dz$$ for each measurable function $g$ where $f_(X_1,X_2)$ denotes the joint density of $X_1$ and $X_2$

Now take $$A = Z < z $$ and consider that for independent random variables $X_1,X_2$ it holds $$f_(X_1,X_2)(x_1,x_2) = f_X_1(x_1)f_X_2(x_2)$$ to get

$$P(Z < z) = int_Bbb R^2 1_aX_1 - X_2 < z, dx_1dx_2$$

Solving the integral will give you the wanted result.

Hint:

For fixed $z$:$$mathsf P(Z<z)=int_0^1int_0^1[ax-y<z]dydx$$ where $[ax-y<z]$ is the function $mathbb R^2tomathbb R$ taking value $1$ if $ax-y<z$ and taking value $0$ otherwise.