The probability density function of $X$, the life time of a certain type of electronic component(measured in hours), is given by $$f(t)=\begin{cases}\frac{10}{x^2}~~~~~~~~~~~~x>10\\ 0, ~~~~~~~~~~~~~x\leq 10\end{cases}$$
$(a).$ Show that $f(x)$ is a legitimate pdf
$(b).$ Find the expected life time of these components
For the first part we have to show $\int_{-\infty}^{\infty}f(x)dx=1 $
For the second part we have to find $E(X)=\int_{-\infty}^{\infty}xf(x)dx$
$$E(X)=\int_{-\infty}^{\infty}xf(x)dx=\int_{10}^{\infty}\frac{10}{x}dx$$
but this is not convergent so I can't find $E(X)$ is there anything wrong what I did? I can't understand why $E(X)$ can't be found
If I am correct please tell me why I can't find the $E(X)$ as I think every probability density function has expected value
