Compute E[X] and P[Y>2]

The joint probability density function of X and Y is given by

$\displaystyle

f(x,y) = \begin{cases} exp[-(x+y)] \qquad 0 \le x < \infty \; , \; 0
\le y < \infty \\
0 \qquad , otherwise
\end{cases}
$

Compute E[X] and P[Y>2]

Quote:

---

Originally Posted by Stats

The joint probability density function of X and Y is given by

$\displaystyle

f(x,y) = \begin{cases} exp[-(x+y)] \qquad 0 \le x < \infty \; , \; 0
\le y < \infty \\
0 \qquad , otherwise
\end{cases}
$

Compute E[X] and P[Y>2]

---

As with most of the questions you have asked, apply the basic definitions:

$\displaystyle E(X) = \int_{0}^{+ \infty} \int_{0}^{+ \infty} x e^{-(x+y)} \, dy \, dx$.

$\displaystyle \Pr(Y > 2) = \int_{2}^{+ \infty} \int_{0}^{+ \infty} e^{-(x+y)} \, dy \, dx$.

Now integrate (it is assumed that you have the necessary calculus background).

By inspection the two rvs are independent.

X is an exponential with mean 1, hence EX=1, same for Y.

Since $\displaystyle \int_0^{\infty} e^{-x} \, dx=1$ you only need to compute $\displaystyle \int_2^{\infty}e^{-y} \, dy =e^{-2}$.