I'm new to this site and was wondering if anyone could help me on a math question!
I am currently studying probability and statistics and have come across a challenging question for me. See link below.
So previous I have worked with joint density functions in order to determine probabilities but I'm afraid I'm totally stuck when dealing with independent density functions.
Could anyone help me on this matter please. Kind regards.
Suppose that the random variable $X$ has density $$ f_X(x) = \begin{cases} \frac{x}{2}, & 0<x<2 \\ 0, & \text{otherwise} \end{cases} $$ and suppose that the random variable $Y$ has density $$ f_Y(y) = \begin{cases} 4y^3, & 0<y<1 \\ 0, & \text{otherwise} \end{cases} $$ also suppose that the variables $X$ and $Y$ are independent.
Determine $P(Y>X)$ to three decimal places.
Easy answer:
$$\mathbb{P}[Y>X]=\int_{0}^{1}f_Y(y)\Bigg[\int_0^y f_X(x)dx\Bigg]dy$$