Question
The joint probability density function of X and Y is given by
f(x,y) = $\displaystyle \begin{cases} exp[-(x+y)] & 0 \le x < \infty , 0 \le y < \infty \\
0 & otherwise \end{cases}$
Compute E[X] and P[Y>2]
Here is what i have done
g(x) = $\displaystyle \int_{0}^{ \infty}$ f(x,y) dy
= $\displaystyle \int_{0}^{ \infty} $ exp[-(x+y)] = -$\displaystyle \int_{0}^{ \infty}$ exp(x+y) dy
= - $\displaystyle \int_{0}^{ \infty}$ $\displaystyle exp(x) \ . \ exp(y) dy$
= -exp(x) $\displaystyle \int_{0}^{ \infty}$ exp(y) dy = ?_________I am stuck here what is exp(y)....
Why are you confused? Matheagle's post says clearly that $\displaystyle g(x) = e^{-x}$. You should be able to calculate $\displaystyle \int_{0}^{ \infty}e^{-y} \, dy = 1$ .... (Besides, a pdf cannot be negative and $\displaystyle -e^{x} < 0$ for all real values of x ....!!)