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This paper proves a probabilistic version of Taylor's theorem \begin{equation*} \mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...

Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables. Distribution of $Z=\dfrac{X}{Y}$ Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...

Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $. We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$. Do you have any ...

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...