# finding cdf of standard uniform

• Nov 23rd 2010, 03:17 PM
chutiya
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?
• Nov 23rd 2010, 03:58 PM
mr fantastic
Quote:

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.
• Nov 23rd 2010, 04:25 PM
chutiya
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?
• Nov 23rd 2010, 04:50 PM
harish21
correct!