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

suppose X has normal distribution with $\mu=\sigma^2=1$. how can calculate $E(\Phi(X-1))$
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### 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 ...
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### 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....
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### 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?
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### 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 ...
645 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 ...
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594 views

### How can I compute $\int F(x \mid a,b)f(x \mid w,z) {}dx$ in closed form?

Suppose $F$ is the cumulative distribution function of the normal distribution with mean $a$ and standard deviation $b$, and suppose $f$ is the probability density function of the normal distribution ...
### How can I calculate this integral? $\int_{-\infty}^{\infty} \Phi (a + bX) \phi (c + eX) dx$
Suppose we have the density and distribution of the standard normal. How can one calculate the integral: $\int_{-\infty}^{\infty} \Phi (a + bX) \phi (c + eX) dx$ Note this is not included in the ...
I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where \$A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^2}},\ ...