482 views

### What is the proof of $\mathbb{E}\Phi (X) = \Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$, where $X \sim \mathcal{N}(\mu,\sigma^2)$? [duplicate]

Let $X \sim \mathcal{N}(\mu,\sigma^2)$. I think it's true that $$\mathbb E \Phi(X) = \Phi \left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$$ where $\Phi$ is the cdf of standard normal. This holds up under ...
• 736
117 views

### how can calculate $E(\Phi(X-1))$ [duplicate]

suppose X has normal distribution with $\mu=\sigma^2=1$. how can calculate $E(\Phi(X-1))$
• 1
142 views

### Closed-form solution for an integral involving the p.d.f. and c.d.f. of a $N(0, 1)$-distributed random variable [duplicate]

Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the probability and cumulative density functions, respectively, of a random variable with distribution $\text{N}(0,\,1)$. I was wondering if you could help me to ...
53 views

### What' s $E[Y] = E[f(X)]$? $X\sim N(\mu,\sigma^2)$ and $f()$ is cdf of std normal rv? [duplicate]

What's the expected value of $Y= \Phi(X)$ where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$ and $\Phi$ being the cdf of a standard normal distribution?
37 views

### Normalizing the constant of the posterior [duplicate]

I am reading the lecture note from Cambridge University about Probabilistic Ranking and they claim that the normalized constant has a closed form in the below formula but I could not know how to prove....
• 153
10k views

### Distribution of the maximum of two correlated normal variables

Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly normal with correlation coefficient $r$. What is the distribution function of $\max(X_1, X_2)$?
• 2,253
983 views

### Distribution of sum of two independent normals conditional on one of them

Assume $X$ and $Y$ are iid $N(0,1)$. I am looking for a "neat" expression for $$P\left(\frac{X+Y}{\sqrt{2}}>c\,\Biggl|\,X<c\right).$$ Related question seem to be discussed here or here, but if ...
• 31.8k
8k views

### Are there applications for differential equations in statistics? [closed]

So I know we statisticians don't use differential equations as heavily as e.g. engineers. Actually, I have never seen or needed them in my studies. I'm curious to learn about them now, and I'd be ...
• 4,281
1 vote
1k views

### Calculating the integral of two normal CDFs with a normal distribution

I'm trying to calculate: $$\int\Phi((x-\mu_{1})/\sigma_{1})*\Phi((x-\mu_{2})/\sigma_{2})*\phi(x)dx$$ where $\Phi$ and $\phi$ are the standard normal cumulative distribution function and probability ...
• 38
2k views

• 143