# Math Help - finding cdf of standard uniform

1. ## finding cdf of standard uniform

If $X \;and\; Y$ are independent $standard \;uniform$ random variables. and we want to find the cumulative distribution of $X-2Y$

to find the cdf here, can I suppose that $W=X-2Y,0 and then find $P(W \leq w)$

so, I have $P(W \leq w) = P(X-2Y \leq W) = P(Y \geq \frac{W-X}{2}) = 1 - P(Y < \frac{W-X}{2})$

and integrate $1 - \displaystyle \int _0^w\;\int_0^{\frac{w-x}{2}}\;dy\;dx$

is this right or am I wrong?

2. Originally Posted by chutiya

If $X \;and\; Y$ are independent $standard \;uniform$ random variables. and we want to find the cumulative distribution of $X-2Y$

to find the cdf here, can I suppose that $W=X-2Y,0 and then find $P(W \leq w)$

so, I have $P(W \leq w) = P(X-2Y \leq W) = P(Y \geq \frac{W-X}{2}) = 1 - P(Y < \frac{W-X}{2})$

and integrate $1 - \displaystyle \int _0^w\;\int_0^{\frac{w-x}{2}}\;dy\;dx$

is this right or am I wrong?
Wrong.

For starters it should be $y < \frac{x - w}{2}$. Note also that the support of W is $-2 \leq w \leq 1$ so you will probably have to consider separate cases, that is, the cdf will be a hybrid function.

3. Thank you. I always struggle with drawing the support. Since $-2 \leq w \leq 1$, we have to calculate the cdf for $X-2Y=W,\;-2 \leq w < -1$ and $-1 \leq w < 0$, $0\leq w< 1$ and then integrate, right?

4. correct!