Question

# Suppose the random variables X and Y have a pdf given byf(x,y)=x+y on 0<x<1, 0<y<1Find F(\frac{1}{4},\frac{1}{2}).

Random variables

Suppose the random variables X and Y have a pdf given by
f(x,y)=x+y on 0 Find F($$\frac{1}{4},\frac{1}{2}).$$

2021-05-18

Given that,
X and Y are two random variables
The joint pdf of them is given by,
$$f(x,y)=\begin{cases}x+y\ \ \ 0<x<1, 0<y<1\\0 \ \ \ \ \ otherwise\end{cases}$$
By definition of cumulative distribution function,
$$F(x,y)=\int_{0}^{x}\int_{0}^{y}(s+t)dtds$$
$$=\int_{0}^{x}s(t)_{0}^{y}+(\frac{t^{2}}{2})_{0}^{y}ds$$
$$=\int_{0}^{x}sy+\frac{y^{2}}{2}ds$$
$$=y(\frac{s^{2}}{2})_{0}^{x}+\frac{y^{2}}{2}(s)_{0}^{x}$$
$$=\frac{yx^{2}}{2}+\frac{xy^{2}}{2}$$
Now, $$F(\frac{1}{4}, \frac{1}{2})=\frac{(\frac{1}{2})(\frac{1}{4})^{2}}{2}+\frac{(\frac{1}{2})^{2}(\frac{1}{4})}{2}=\frac{3}{64}$$