Joint pdf of X and Y with absolute value range

Question

I have the following joint pdf:

$f(x,y)=0.5$ where $0 \leq|x|\leq|y|$, $0 \leq|y|\leq1$, and $0$ otherwise

The question is: are $X$ and $Y$ independent and uncorrelated?

I know that if $f(x)$*$f(y)$=$f(x,y)$, then $X$ and $Y$ are independent, and $Cov(x,y)$=$0$ means uncorrelated.

I found $f(x)$ using integral $0.5dy$ range($x$ to 1) + $0.5dy$ range(-1 to $x$) = 1 and

$f(y)$ using integral $0.5dx$ range($y$ to 0) + $0.5dx$ range(0 to $x$) = 0

So $1*0$ is not $0.5$, means $X$ and $Y$ are not independent and correlated.

But I have doubt in choosing the ranges of integral because of absolute value. Can anybody help with this?

Answer 1

First off, always draw a picture.

We certainly could do this problem by integration to get the marginal distributions $f_X$ and $f_Y$. Your integrals are not quite right, however, because of some sign ambiguities.

We have to set up the integrals to cover the right areas. It's easier to keep the limits in terms of the absolute values. For $y$ there is one interval where $x$ ranges from $-|y|$ to $|y|$. For $x$ there are two intervals, and hence two integrals. Abusing notation slightly, but hopefully in a clear way:

$$ \begin{align*} f_X(x) &= \int f(x,y) \text{ d}y = \int_{|x| < |y|} 0.5 \text{ d}y = \int_{y = -1}^{y = -|x|} 0.5 \text{ d}y + \int_{y = |x|}^{y = 1} 0.5 \text{ d}y= 1 - |x| \\ f_Y(y) &= \int f(x,y) \text{ d}x = \int_{|x| < |y|} 0.5 \text{ d}x = \int_{x=-|y|}^{x=|y|} 0.5 \text{ d}x = |y| \end{align*} $$

$f_X(x) f_Y(y) = (1 - |x|)|y| \neq f(x,y)$, so they are not independent.

However, we don't need to do the integrals to find this -- the condition that $f(x,y) = f_X(x) f_Y(y)$ means that slices of $f(x,y)$ at constant $x$ values must be scalar multiples of each other. By inspection this is not true: $f(0.5, y)$ is not a multiple of $f(0.6, y)$.

The covariance between x and y can be done by setting up integrals, but it can also readily be seen by symmetry to be 0.

Answer 2

Once you realize that one should (always) draw a picture, the following might help:

Grand Principle 1: Include the conditions on the ranges in the densities as indicator functions.

In your case, the density $f_{(X,Y)}$ of $(X,Y)$ is defined for every $(x,y)$ in $\mathbb R\times\mathbb R$ by $$ f_{(X,Y)}(x,y)=\frac12\mathbf 1_{0\le |x|\le |y|}\mathbf 1_{0\le |y|\le 1}, $$ and the question is to know whether there exists some functions $g$ and $h$ defined on $\mathbb R$ such that, for every $(x,y)$ in $\mathbb R\times\mathbb R$, $$ f_{(X,Y)}(x,y)=g(x)h(y). $$ If this is so, you know that (1) $g$ is a multiple of the density of $X$, (2) $h$ is a multiple of the density of $Y$, and (3) $X$ and $Y$ are independent. Otherwise, $X$ and $Y$ are not independent.

Second remark:

Grand Principle 2: A way to disprove independence is to find a product $B_1\times B_2$ such that $f_{(X,Y)}$ is zero on $B_1\times B_2$ but neither zero on $B_1\times\mathbb R$ nor zero on $\mathbb R\times B_2$.

In other words, one is looking for a zone where $f_{(X,Y)}$ is zero but ought not to.

In your case, $B_1=(\frac12,1)$ and $B_2=(0,\frac12)$ would do. To see this, note that $f_{(X,Y)}$ is zero on $B_1\times B_2$ but that $B_1\times\mathbb R\supset(\frac12,\frac34)\times(\frac34,1)$ and $\mathbb R\times B_2\supset(0,\frac14)\times(\frac14,\frac12)$, which both have positive probability. Hence you are done.