# Solving probability from joint PDF

What am I doing wrong?

My thinking for solving this probability: First I attain the marginal probability density of $f(x,y) \rightarrow \ f_x(x)$ Then attain the integral. However I am receiving a value greater than 1.

$$f(x,y) = \begin{cases} 4x^3, & \text{0 < y < x < 100} \\ 0, & \text{otherwise} \end{cases}$$

$$P(x>2) = \int_{2}^{100} f_x dx$$

where $f_x$ is the marginal probability density for the joint probability density

• I think something's wrong with the joint pdf, as it doesn't seem to have total integral $1$. – carmichael561 Oct 13 '16 at 5:10
• @carmichael561 Is my methodology correct though? – user Oct 13 '16 at 5:11

## 1 Answer

There is nothing wrong with your plan.   That is just what you need to do.

$$\mathbb P(X>2) =\int_2^{100}\color{blue}{\underbrace{\bbox[lemonchiffon,0.5ex]{\color{black}{\int_0^x f(x,y)\operatorname d y}}}_{f_X(x)}}\operatorname d x$$

The only issue is that you do not have a valid probability density function.

$$\int_0^{100}\int_0^x 4x^3\operatorname d y\operatorname d x \neq 1$$