Here's the problem.
We have a random variable X that follows a Poisson law. If we take the log of this variable, what are the first two moments (mean and variance) of the law it follows?
This looks like a simple question, but I can't find anything about it. Any idea?
EDIT: In order to prevent the X=0 case, we bias the Poisson law. Our random variable becomes log(X+epsilon) with X~Poisson(λ).
Best Answer
Recall that the mean $\mathbb{E}(X)$ of a random variable $X$ is a weighted average of the possible values of $X$, weighted by the probability $P(X = x)$ of each outcome $x$ of $X$. Hence, the mean of a function $f(X)$ is a weighted average of the possible values of $f(X)$, weighted by the probability $P(f(X) = f(x))$. But the probability that $f(X)$ takes the value $f(x)$ is the same as the probability that $X$ takes the value $x$ (this is not exactly true, since $f$ might take the same value for multiple $x$, but for the purpose of taking a weighted average it doesn't matter since the effective weight on $f(x)$ will be the sum of the probabilities of each such $x$ either way). This allows us to use the following formula to find the mean of $f(X)$, with sums or integrals depending on whether $X$ is continuous or discrete: $$\mathbb{E}(f(X)) = \int f(x) \cdot P(X = x) dx$$ $$\mathbb{E}(f(X)) = \sum f(x) \cdot P(X = x).$$
Similarly, the variance is a weighted average of the squared difference between each value $x$ and the average value $\mathbb{E}(X)$, weighted by the probability $P(X=x)$. So we can similarly calculate the variance of $f(X)$ "by hand" once we have the mean, using: $$\text{Var}(f(X)) = \int \left(f(x) - \mathbb{E}(f(X)) \right)^2 \cdot P(X = x) dx$$ $$\text{Var}(f(X)) = \sum \left(f(x) - \mathbb{E}(f(X)) \right)^2 \cdot P(X = x).$$
In both cases, the sum or integral is taken over all possible values $x$ of your variable $X$.
Note that as per Henry's answer, naively plugging $f(x) = \log(x)$ into these formulas will get you in trouble if $X \sim \text{Poisson}(\lambda)$, as you'll have a diverging integral. For a Poisson distribution, which allows $X = 0$, taking the mean and variance of $log(X)$ is only sensible if you first exclude $X = 0$ from your domain somehow. So this answer is conditional on $X$ remaining in the domain of $f$.