If \(X\) is follows a standard normal distribution, what is the expected value of \(\mathbb{E}(e^X)\)?

Given a uniform random variable \(U\), how to create a random variable with distribution function \(F\)?

Let X be a positive continuous random variable. Prove that \(\mathbb{E}X=\int_{0}^{\infty}\mathbb{P}(X\geq x)dx\)

Let X be a continuous random variable with PDF

\(f(x)=\begin{cases}4x^3&0<x\leq1\\0&otherwise\end{cases}\)

Calculate \(\mathbb{P}\left(X\leq \frac{2}{3}|X> \frac{1}{3}\right)\)

\(f(x)= \begin{cases} x^2\left(2x+\frac{3}{2}\right)&0<x\leq 1\\ 0&otherwise \end{cases}\)

If \(Y=\frac{3}{X}+5\), Calculate \(Var(Y)\)

Let X be a random variable with PDF given by

\(f(x)= \begin{cases} cx^2&|x|\leq 1\\ 0 & otherwise \end{cases}\)

(a). Find constant c

(b). Calculate \(\mathbb{P}\left(X>\frac{1}{2}\right)\)

(c). Calculate E(X) and Var(X)

Let X be a continuous random variable with PDF given by \(f(x)=\frac{1}{2}e^{-|x|}\) If \(Y=X^2 \), find the cdf of \(Y\)