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

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 $$P(X+Y\leq 6)$$

## Expert 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