Linked Questions
28 questions linked to/from How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$
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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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answer
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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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3
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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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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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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
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votes
2
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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 $\max(X_1, X_2)$?
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2
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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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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 ...
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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 ...
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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 $\rho \approx 0.3$.
$$
\begin{pmatrix}
X_1 \\
X_2
\end{pmatrix} \sim \mathcal{N} \left( \...
- 153
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votes
1
answer
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Variance of a Cumulative Distribution Function of Normal Distribution
Suppose, $X\sim N(\mu,\sigma^2)$.
Can anyone help in finding the following : $\text{Var } \bigg(\Phi\big(\frac{X + c}{d}\big) \bigg)$ ? Here, c and d are positive.
Here, $\Phi(x)$ is the "...
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2
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Let $X_{(1)}\leq X_{(2)}$ be the order statistics. Evaluate $\operatorname{Var}(X_{(j)})$, $\operatorname{Cov}(X_{(1)},X_{(2)})$
Let $X_{(1)}\leq X_{(2)}$ be the order statistics for a random sample of size $2$ from a normal distribution with mean $\mu$ and variance $\sigma ^{2}$.
Evaluate $\operatorname{E}(X_{(1)})$, $\...
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How can I compute $\int_{-\infty}^{\infty}\Phi\left(az+b\right)^{2}\phi(z)\,dz$ in closed form?
How can one evaluate the expectation of the squared normal CDF in closed-form?
$$\mathbb{E}\left[\Phi\left(aZ+b\right)^{2}\right] = \int_{-\infty}^{\infty}\Phi\left(az+b\right)^{2}\phi(z)\,dz$$
Here,...
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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<...
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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 ...
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