# Tagged Questions

Questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution.

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### Error function relation to the normal cumulative distribution function

A CDF for a normal standard is the following: $$N(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\phi^2/2} d\phi$$ I have the following relation in my notes which I am not very sure how they ...
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### Maximum and minimum value of $P(-4<Y<6)$, where Y has normal distribution with standard deviation 2 and the mean unknown.

Let Y be a random variable has normal distribution with standard deviation 2 and mean is unknown. Find the maximum and minimum value of $P(-4<Y<6)$.
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### T-Distribution, Normal Distribution, and Confidence Intervals

In my probability class we were given the following problem: Suppose you take a sample of your friends and measure their heights. You calculate the sample mean to be 5 feet tall and the sample ...
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### Shape of chi-square distribution df=1

I am trying to understand, intuitively, the shape of the chi-square distribution with 1 degree of freedom. Let $X$ be a random variable whose distribution is given by the standard normal distribution....
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### Prove $\lim_{\Delta t \to 0} \frac{2}{\Delta t} \left(1-\Phi\left(\frac{\epsilon}{\sqrt{\Delta t}}\right)\right) = 0$

Given $$\lim_{\Delta t \to 0} \frac{2}{\Delta t} \left(1-\Phi\left(\frac{\epsilon}{\sqrt{\Delta t}}\right)\right)$$ with $\Phi$ standard normal CDF, how can I prove the limit to be equal to $0$? ...
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### get data to draw a gauss curve

I would like to know how to get some data from a normal distribution to draw its gauss curve. I have the standard deviation, the average and the x, but I don't know how to get some points to draw the ...
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### Joint distribution of multivariate normal distribution

So the question asks: Let $X = (X_1, ... ,X_{2n})$~ $N (0, ∑)$ (multivariate normal distribution with mean vector $(0,..., 0)$ and covariance matrix $∑$ ), where $n≥ 1$. Find the joint distribution ...
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### What is the probability that K out of N normal random variates have the same sign?

Assume we have a Multivariate Normal distribution. For simplicity, let all N random variables have a zero mean and unit variance. Also for simplicity, let the correlation between all pairs of random ...
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### Integral of product of two normal distribution densities

I want to compute the integral: $\displaystyle \int^{\infty} _{-\infty} \frac{1}{\sqrt{2\pi}} e^{-\frac{(y-x)^2}{2}} \frac{1}{\sqrt{2\pi}ab} e^{-\frac{x^2}{2(ab)^2}} dx$ Maybe we can use that ...
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### Product Distribution and Expectation

Let $X_1, \ldots, X_d$ be $d$ independent Gaussian $N(0,1)$ random variables, and let $$Y=\frac{1}{\| X \|} (X_1, \ldots ,X_d)$$ Clearly $Y$ lies on the surface of the sphere $S^{d-1}$. Let the ...
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### Prove that $E(e^{sX^2}) = \dfrac{1}{\sqrt{1-2s}}$

If $X$ is normally distributed with mean $0$ and SD $1$, show that $$E(e^{sX^2}) = \dfrac{1}{\sqrt{1-2s}}$$ for $s < \dfrac{1}{2}$. I obtain this from the paper 'Elementary proof of Johnson and ...
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### If $B_t - B_s, \ 0\leq s < t,$ is normally distributed, there are constants $C_n, \ E|B_t - B_s|^{2n}=C_n|t-s|^n$

I am working on the following problem: Show that if $B_t - B_s, 0 \leq s < t,$ is normally distributed with mean zero and variance $t-s$, then for each positive integer $n$ there is a positive ...
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### What are the distributions of these Gaussian variables?

I'm just wondering if I have the correct answers to these questions. Let $X$, $Y$, and $Z$ be multivariate Gaussian distributed with mean vector and covariance matrix:  \mu = (0,1,2)^T, \quad \...
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### How to derive the expected value of $(X^HX)^{-1}X^H\mathrm{diag}(XX^H)X(X^HX)^{-1}$ when $X$ is the complex Gaussian matrix?

Let $X$ be the $N\times K$ matrix, of which elements are independent identically distributed zero-mean complex Gaussian random variables with unit variance. Denote $Y =X(X^HX)^{-1}$ as the pseudo-...
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### Finding Mean and Distribution of Normal Random Variables

Assume that $X_1$, and $X_2$ are i.i.d. normal random variables with mean $0$ and variance $1$. Let $Y_1$ and $Y_2$ be defined as $Y_1 =8X_1+6X_2$ and $Y_2 = X_1$. $E[Y_1]= 0$ correct? because it'...
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### Sample mean converging to normal much faster than expected.

I am taking a non-normal distribution (Poisson, Exponential or Uniform etc.) and I simulate thousands of experiments for small sample sizes ($n=1,...,10$). I calculate the 95%-confidence interval each ...
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### How to solve $P(X=a) = P(X=b)$

A random variable X is normally distributed with $\mu = 60$ and $\sigma$ = 3. What is the value of 2 numbers a,b so that $P(X=a) = P(X=b)$. The solution is $a = 60$ and $b = 65$. However, I do not ...
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### Gaussian Distribution in the form of rayleigh and uniform

I have this form $x=r \cos (\phi)$, where $r$ is Rayleigh distributed RV, $r$ ~ $Ray (\sigma_r)$ and $\phi$ is uniform in $[0,2\pi]$. Does $x$ follow a Gaussian distribution? Can you please provide ...
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### What is the expected distance from the mean of a multivariate Gaussian?

For a multivariate Gaussian distribution $p(x) = N(x\mid \mu,\Sigma)$, what is $E[\|x-\mu\|]$? I know from this question that $E[|x-\mu|]=\sigma\sqrt{2/\pi}$ for univariate Gaussians. But I couldn'...
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### Getting a Number that Doesn't Make Sense to Me

Information about question: the duration of a movie trailer is approximately normal, with mean 150 seconds and standard deviation 30 seconds. One part in the questions is not making sense to me and I'...
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### Random walk in high dimensional space with stationarity

I have a vector of high dimension ( say 100). When I take a random walk ( i.e add a step value to each components of the vector, the step value being drawn randomly drawn from standard normal ...
Let's say that $Y = \log T = \alpha + \sigma W$. I know that If $W$ has logsitic distribution, the $T$ will have the log-logistic distribution. Also, if $W$ has the standard normal distribution then \$...