Question

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+Y>3)

Random variables
ANSWERED
asked 2021-05-23

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

Answers (1)

2021-05-24

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 \(P(X + Y > 3)\) is obtained as 0.1991 from the calculation given below:
\(P(X+Y>3)=1-P(X+Y\leq 3)\)
\(=1-\int_{0}^{3}\int_{0}^{3-x}e^{-(x+y)}dxdy\)
\(=1-\int_{0}^{3}\int_{0}^{3-x}e^{-x}e^{-y}dxdy\)
\(=1-\int_{0}^{3}e^{-x}dx\int_{0}^{3-x}e^{-y}dy\)
\(=1-\int_{0}^{3}e^{-x}dx[-e^{-y}]_{0}^{3x}\)
\(=1-(-\int_{0}^{3}e^{-x}dx[e^{-(3-x)}-e^{-0}])\)
\(=1-(-\int_{0}^{3}e^{-x}dx[e^{-3+x}-1])\)
\(=1-\int_{0}^{3}e^{-x}dx[1-e^{-3+x}]\)
\(=1-\int_{0}^{3}e^{-x}-e^{-3}e^{x}e^{-x}dx\)
\(=1-([-e^{-x}]_{0}^{3}-e^{-3}[x]_{0}^{3})\)
\(=1-(-[e^{-3}-e^{-0}]-e^{-3}[3-0])\)
\(=1-([1-e^{-3}]-3e^{-3})\)
\(=1-1+e^{-3}+3e^{-3}\)
\(=4e^{-3}\)
=0.1991
Thus, the probability of \(P(X + Y > 3)\) is 0.1991.

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