# Suppose that X and Y are continuous random variables with joint pdf f(x,y)=e^{-(x+y)} 0<x<\infty\ and\ 0<y<\infty and zero otherwise.Find P(X>3)

Suppose that X and Y are continuous random variables with joint pdf $$f(x,y)=e^{-(x+y)} 0$$ and zero otherwise.
Find $$P(X>3)$$

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Arnold Odonnell

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.