Random variable $x$ follows normal distribution and we know the mean value of $x$. Is there a well-known way to compute the mean value of $\frac{1}{x}$? Perhaps, some way to estimate the integral of $pdf(x)*(\frac{1}{x})$ from zero to infinity?
Asked
Active
Viewed 76 times
0
-
1The mean of $1/X$ does not exist in your case. Consider $X \sim N(0, 1)$... – JACKY88 Jan 23 '13 at 22:17
1 Answers
1
Generally if you know the pdf of the random variable X, try to write the pdf of 1/X then compute the mean. The computation can be eased if you know that X is a positive random variable.
In the case of normal, its tricky because 1/X is not defined for X = 0, so you cannot even discuss the random variable 1/X to begin with before you compute the mean.
If we know X is always positive
$F_{1/X}(t) = P(t\le1/X) = P(X \le 1/t) = F_X(1/t)$ for t positive
$F_{1/X}(t) = P(t\le1/X) = 1 - P(X \le 1/t) = 1 - F_X(1/t)$ for t negative
then you have CDF $F_{1/X}(t)$, therefore can compute PDF $f_{1/X}(t)$
mez
- 10,497