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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

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

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