Question

Random variables X and Y have joint PDF f_{X,Y}(x,y)=\begin{cases}12e^{-(3x+4y)},\ x \geq 0, y \geq 0\\0,\ otherwise\end{cases} Find P[X+Y\leq 1]

Random variables
ANSWERED
asked 2021-05-28
Random variables X and Y have joint PDF
\(f_{X,Y}(x,y)=\begin{cases}12e^{-(3x+4y)},\ x \geq 0, y \geq 0\\0,\ otherwise\end{cases}\)
Find \(P[X+Y\leq 1]\)

Answers (1)

2021-05-29

The value of \(P[X+Y\leq 1]\) is obtained as given below:
\(f(x,y)=12e^{-(3x+4y)}. x\geq 0 \text{ and } y \geq 0\)
\(f(x)=\int_{y}f(x,y)dy\)
\(=\int_{0}^{\infty}12e^{-(3x+4y)}dy\)
\(=12e^{-3x}[-\frac{e^{-4y}}{4}]_{0}^{\infty}\)
\(=12e^{-3x}\times \frac{1}{4}\)
\(f(x)=3e^{-3x}\)
\(P(X+Y\leq 1)=P(X\leq 1-Y)\)
\(=\int_{0}^{1-y}f(x)dx\)
\(=\int_{0}^{1-y}3e^{-3x}dx\)
\(=3[-\frac{e^{-3x}}{3}]_{0}^{1-y}\)
\(=-[e^{-3(1-y)}-1]\)
\(=[1-e^{-3(1-y)}]\)

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