Step 1

Introduction:

The joint density function of two random variables X and Y is given below:

\(f(x,y)=\left\{\begin{matrix}e^{-(x+y)}.\ \ \ 0\leq x \leq \infty, 0\leq y < \infty \\0.\ elsewhere \end{matrix}\right\}\)

The marginal density function of X is,

\(f(x)=\int_{0}^{\infty}e^{-(x+y)}dy\)

\(=e^{-x}\int_{0}^{\infty}e^{-y}dy\)

\(=-e^{-x}[-e^{-y}]_{0}^{\infty}\)

\(=-e^{-x}[e^{-\infty}-e^{-0}]\)

\(=-e^{-x}[0,1]\)

\(=e^{-x}\)

Step 2

The probability of \(P(X > 3)\) is obtained as 0.0498 from the calculation given below:

\(P(X>3)=\int_{3}^{\infty}e^{-x}dx\)

\(=[-e^{-x}]_{3}^{\infty}\)

\(=-[e^{-\infty}-e^{-3}]\)

\(=-[0-e^{-3}]\)

\(=e^{-3}\)

=0.0498

Thus, the probability of \(P(X > 3)\) is 0.0498.