Question

The joint density function of two continuous random variables X and Y is:f(x,y)=f(x,y)=\begin{cases}cx^{2}e^{\frac{-y}{3}} \& 1<x<6, 2<y<4 = 0\\0 & otherwise\end{cases}

Random variables
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asked 2021-06-05

The joint density function of two continuous random variables X and Y is:
\(f(x,y)=f(x,y)=\begin{cases}cx^{2}e^{\frac{-y}{3}} & 1<x<6, 2<y<4 = 0\\0 & otherwise\end{cases}\)
Draw the integration boundaries and write the integration only for \(P(X+Y\leq 6)\)

Answers (1)

2021-06-06

Integration boundaries for \(X+Y\leq 6\):
X: 1 to 6-y
Y: 2 to 4
Integration:
\(P(X+Y\leq 6)=\int_{y=2}^{4}\int_{x=1}^{6-y}0.0186x^{2}e^{\frac{-y}{3}}dxdy\)
\(=0.0186\int_{y=2}^{4}e^{\frac{-y}{3}}[\frac{x^{3}}{3}]_{1}^{6-y}dy\)
\(=0.0186\int_{2}^{4}e^{\frac{-y}{3}}[\frac{(6-y)^{3}}{3}-\frac{1}{3}]dy\)
\(=0.0062\int_{2}^{4}e^{\frac{-y}{3}}[(6-y)^{3}-1]dy\)
\(=0.0062\cdot24.0489\)
=0.1491

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