What is the proof of $\mathbb{E}\Phi (X) = \Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$, where $X \sim \mathcal{N}(\mu,\sigma^2)$?
177
How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$
27347
Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.
108560
Proof of a Gaussian Integral property
36384
Q: $\lim_{n\to \infty}\left(1 + \frac{1}{n}\right)^{n} = e$
12976
Prove $\displaystyle\lim_{n\to\infty}\phi(n)=\infty$.
30431
How is the notation $X\sim N(\mu,\sigma^2)$ read?
63218
Is conditional expectation with respect to two sigma algebra exchangeable?
132392
Properties of Expectation
67489
How to show that $E[(\hat\theta -\theta)^2]<Var(\bar X)=\dfrac{1}{n}$?
138819
Prove that $\phi(n) \geq \sqrt{n}/2$
156901
Showing that $X\sim E[X\mid\mathcal{G}]$ implies $X=E[X\mid\mathcal{G}]$ almost surely
7384
How do I prove that $m(TE)=\left|\det(T)\right|m(E)$?
140790
Prove that $E(X) = \int_{0}^{\infty} P(X>x)\,dx = \int_{0}^{\infty} (1-F_X(x))\,dx$.
4287
Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
118253
Conditional expectation of $X$ given $\sin(X)$
151157
Justification of proof of law of total expectation
134696
How can I show that the conditional expectation $E(X\mid X)=X$?