Linked Questions

5 votes
0 answers

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 ...
xiaodai's user avatar
  • 736
0 votes
1 answer

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

suppose X has normal distribution with $\mu=\sigma^2=1$. how can calculate $E(\Phi(X-1))$
hadi's user avatar
  • 1
3 votes
0 answers

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 ...
Rodrigo Rodriguez's user avatar
0 votes
0 answers

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?
Clover_1621's user avatar
2 votes
0 answers

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....
tndoan's user avatar
  • 153
24 votes
2 answers

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 $\max(X_1, X_2)$?
CuriousMind's user avatar
  • 2,253
6 votes
2 answers

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 ...
Christoph Hanck's user avatar
5 votes
0 answers

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 ...
Alexander Engelhardt's user avatar
1 vote
2 answers

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 ...
cfen's user avatar
  • 38
5 votes
1 answer

How to calculate the total probability inside a slice of a bivariate normal distribution in R?

I have a bivariate normal distribution composed of the univariate normal distributions $X_1$ and $X_2$ with $\rho \approx 0.3$. $$ \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim \mathcal{N} \left( \...
Michael Clark's user avatar
9 votes
1 answer

Variance of a Cumulative Distribution Function of Normal Distribution

Suppose, $X\sim N(\mu,\sigma^2)$. Can anyone help in finding the following : $\text{Var } \bigg(\Phi\big(\frac{X + c}{d}\big) \bigg)$ ? Here, c and d are positive. Here, $\Phi(x)$ is the "...
Dwaipayan Gupta's user avatar
9 votes
2 answers

Let $X_{(1)}\leq X_{(2)}$ be the order statistics. Evaluate $\operatorname{Var}(X_{(j)})$, $\operatorname{Cov}(X_{(1)},X_{(2)})$

Let $X_{(1)}\leq X_{(2)}$ be the order statistics for a random sample of size $2$ from a normal distribution with mean $\mu$ and variance $\sigma ^{2}$. Evaluate $\operatorname{E}(X_{(1)})$, $\...
Diego Fonseca's user avatar
13 votes
1 answer

How can I compute $\int_{-\infty}^{\infty}\Phi\left(az+b\right)^{2}\phi(z)\,dz$ in closed form?

How can one evaluate the expectation of the squared normal CDF in closed-form? $$\mathbb{E}\left[\Phi\left(aZ+b\right)^{2}\right] = \int_{-\infty}^{\infty}\Phi\left(az+b\right)^{2}\phi(z)\,dz$$ Here,...
Andrei's user avatar
  • 143
3 votes
1 answer

Conditional distribution of a normal distribution given it is smaller/bigger than another normal distribution

Say I have two independent random variables $X \sim N(u_1, \sigma_1)$ and $Y \sim N(u_2, \sigma_2)$. I want to get the conditional distribution of X given whether X is bigger than Y or not. $P(X|X<...
HannaMao's user avatar
2 votes
1 answer

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 ...
SG2013's user avatar
  • 31

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