I am given a joint distribution of two random variables, X and Y, which is as follows:

$$ f(x,y)= 8xy \\ 0≤y≤x≤1 $$

Now I need to find :

$$ P[X\leq 1/2] $$

Now I now I need to find marginal pdf first but I am confused between the limits. Will the limits be from $0$ to $1/2$, since we need to substitute the limits of y, if not, then what is the trick behind the selection of limits if we are given of this type?


That the region common to the support $\{(x,y):0\le y\le x\le 1\}$ of the joint distribution and the restriction $x<1/2$ is $\{(x,y):0\le y\le x\le 1/2\}$ should be clear if you draw a picture.

Then it follows that

\begin{align} P\left(X\le \frac{1}{2}\right)&=E\,[\mathbf1_{X\le1/2}] \\&=\iint \mathbf1_{x\le1/2}\,f(x,y)\,\mathrm{d}x\,\mathrm{d}y \\&=\iint \mathbf1_{x\le 1/2}\,8xy\,\mathbf1_{0\le y\le x\le 1}\,\mathrm{d}x\,\mathrm{d}y \\&=8\iint xy\,\mathbf1_{0\le y\le x\le 1/2}\,\mathrm{d}x\,\mathrm{d}y \end{align}

You are supposed to write the double integral as an iterated integral using Fubini's theorem, and depending on the order of integration, you have

$$\iint xy\,\mathbf1_{0\le y\le x\le 1/2}\,\mathrm{d}x\,\mathrm{d}y=\int_0^{1/2} y\left(\int_y^{1/2} x\,\mathrm{d}x\right)\mathrm{d}y=\int_0^{1/2} x\left(\int_0^x y\,\mathrm{d}y\right)\mathrm{d}x$$

Since the ranges of $x$ and $y$ are dependent, the inner integral is a function of the variable with respect to which you are evaluating the outer integral, while the outer integral runs free of the other variable, giving you a real value as the final answer.

  • $\begingroup$ then the limits will be $0$ to $1/2$ for outer integral and $y$ to $1/2$ for inner? $\endgroup$ Jan 11 '19 at 17:09
  • $\begingroup$ @AhmadQayyum Yes. $\endgroup$ Jan 11 '19 at 18:01

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