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>3)

Random variables
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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 \(P(X>3)\)

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 \(P(X > 3)\) 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.

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