# Find the expected life time of these components

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

• Note every distribution has an expected value (see the Cauchy distribution) – 5201314 Jan 9 at 18:07
• math.stackexchange.com/questions/239288/… – 5201314 Jan 9 at 18:15
• @5201314 thank you got it – puka Jan 9 at 18:38