Given :
\(f(x,y)=\begin{cases}kx(x-y),\\0\end{cases}\)
To find k
\(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)dxdy=1\)
\(=\int_{0}^{2}\int_{-x}^{x}kx(x-y)dxdy=1\)
\(=\int_{0}^{2}\int_{-x}^{x}k(x^{2}-xy)dydx=1\)
\(=\int_{0}^{2}k[x^{2}y-\frac{xy^{2}}{2}]_{-x}^{x}dx=1\)
\(=\int_{0}^{2}k[x^{3}-\frac{x^{3}}{2}+x^{3}+\frac{x^{3}}{2}]dx=1\)
\(=\int_{0}^{2}k[2x^{3}]dx=1\)
\(=2k[\frac{x^{4}}{4}]_{0}^{2}=1\)
=2k(4)=1
\(=k=\frac{1}{8}\)
Suppose that the random variables X and Y have joint p.d.f.
\((f(x,y)=\begin{cases}kx(x-y)\\0\end{cases}\)Find the marginal p.d.f. of the two random variables.
Consider two continuous random variables X and Y with joint density function
\(f(x,y)=\begin{cases}x+y\ o \leq x \leq 1, 0 \leq y \leq 1\\0 \ \ \ \ otherwise\end{cases}\)
\(P(X>0.8, Y>0.8)\) is?
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 \(P(X+Y\leq 6)\)
