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### Expected value of E(exp(1/X)) where X~N(0,sigma_x) [duplicate]

I would like to obtain the expected value of $\mathbb{E}(\exp(1/X))$ where $X$ ~ $N(0,\sigma_x)$
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### For some RV's $X$ with continuous density on $[0,1]$, is $\mathbb{E}|1/X|=\infty$? [duplicate]

Recently I tried to give an argument for why, if $Z \sim N(0,1)$ denotes the standard normal, $\mathbb{E}|\frac{1}{Z}| = \infty$. However, in hindsight this argument does not seem to rely on many ...
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### Mean of inverse exponential distribution

Given a random variable $Y = Exp(\lambda)$, what is the mean and variance of $G=\dfrac{1}{Y}$ ? I look at the Inverse Gamma Distribution, but the mean and variance are only defined for $\alpha>1$ ...
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### Expected value of product of dependent random variables

$X_1$ and $X_2$ are two independent random variables whose expected values are known. I am trying to find the expected value of $(X_1/(X_1+X_2))$. Since $X_1$ and $X_1+X_2$ are dependent, I tried to ...
### Expression for $\mathbb{E}(\exp( ab X Y))$ when $X \sim \text{N}(0, \sigma_X^2)$ and $Y \sim \text{N}(0, \sigma_Y^2)$
Let $X \sim \text{N}(0, \sigma_X^2)$ and $Y \sim \text{N}(0, \sigma_Y^2)$ be independent normal random variables with zero mean, but (possibly) different variances. Given some constants $a$ and $b$, ...