# Tagged Questions

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

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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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### integral of heteroskedastic Gaussian

For a Bayesian analysis I need to solve several integrals of the following kind. Let's start with the simplest 1-D form:  \mathcal{I}_k = \int_{-\infty}^\infty s^k \mathcal{N}\left(s|x,\sigma^2(s)\...
### Find $P(Z>1.8)$ and $P(X>4)$ when $Z$ is normal RV and $X$ is a binomial RV.
Question 1 If $Z\sim N(0,1)$, Find a) $P(Z>1.8)$ b) $P(Z >-2.46)$ c) $P(Z>2.589)$ d) $P(Z<-0.725)$ e) $P(Z<-1.63)$ f) $P(Z>-0.65)$ My Solutions: $P(Z>1.8) = 0.9461$...
So the textbooks says: Let $X_r$ ~ $NB(r,p)$. We could use the probability generating functions to prove that $G_{X_r} (s) = (\frac{ps}{1-(1-p)s})^r$: Let $X$ have the Geometric distribution ...