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

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1answer
18 views

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 ...
1
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2answers
29 views

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 $P(-4<Y<6)$.
0
votes
1answer
40 views

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 ...
0
votes
1answer
42 views

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....
3
votes
1answer
37 views

Probability that a Wiener process is negative at 2 given that it was positive at 1

Let $W_t$ be a standard Wiener process, i.e., with $W_0=0$. If $W_1>0$, what is the probability that $W_2<0$? This is my attempt: we want to determine the conditional probability $$\mathbb P(...
0
votes
0answers
15 views

Relationship between CDF of two normal distributed variables?

If F(X) is cumulative density function of normal distribution of mean 0 and standard deviation of 1, then what is its relationship with F(kX). Here k is constant. Can we express it as g(k)*F(X)?
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2answers
15 views

Confidence Itervals; $Z_{\alpha}$ & $Z_{\alpha/2}$

I'm confused about what exactly $Z_{\alpha}$ is, does there exist a formula for it in terms of $\alpha$? IF so, is there also one for $Z_{\alpha/2}$?
0
votes
1answer
23 views

Gaussian expected inner product with respect to fixed matrix

Suppose $R\in\mathbb{R}^{p\times p}$ is a a fixed matrix (it can be asymmetric, non-positive definite, and so on). Then, I would like to find a formula for the expectation $$\mathbb{E}_{x\sim N_p(0,V)...
0
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1answer
30 views

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 $0$? ...
1
vote
2answers
34 views

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 ...
1
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1answer
46 views

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 ...
0
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0answers
22 views

gaussian distribution. Proakis book

I have this problem from Proakis book: Let $X$ denote a Gaussian random variable with mean equal to zero and variance equal to $\sigma^2$ . Show that: If $n=2k+1 \rightarrow E[X^n] = 0$ else $E[X^n] = ...
-1
votes
1answer
21 views

Z-Score Question Involving Normal Distribution [closed]

The mean of 12 teachers in a school is 38 years with a standard deviation of 5.3. Ms. Data is 24 years old and Mr. Management is 53 years of age. Calculate the z-score of Ms. Data and the ...
1
vote
1answer
42 views

normal reverse hazard rate

Let $$h(x)=\frac{\phi(x)}{\Phi(x)}$$ where $\phi$ and $\Phi$ are PDF and CDF of standard normal distribution. Can we tell that $$h'(x)\geq -1$$ for all $x\in (-\infty,+\infty)$ and $$x+h(x)\geq 0$...
0
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0answers
16 views

PDF of correlated lognormal RVs

Let's say that I have two correlated lognormal RV's, $X$ and $Y$ with $E(X) = 20, SD(X) = 10, E(Y) =15$, and $SD(Y) =5$, where $X$ and $Y$ have correlation coefficient $\rho=.6$. How can I find the ...
1
vote
2answers
71 views

Distribution of a Gaussian variable with a normally distributed mean

Let $X\sim N(0,1)$ and $Y\sim(X,1)$, where $Y-X$ is independent of $X$. Then what is the PDF of $Y$? Specifically, I am interested in computing $P(Y<0\vert X>0)$. For those interested in ...
0
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0answers
18 views

Drawing Contours of Normal Distribution

I have two Classes modeled by normal distributions, with the following mean and covariance matrices. mean1 = \begin{bmatrix}2\\2\end{bmatrix} mean2 = \begin{bmatrix}6\\2\end{bmatrix} cov1 = \begin{...
0
votes
2answers
31 views

Central moments versus moments of a normal random variable

Is this statement correct: $\mathbb{E}[(X_1-\mu)^4]$ of $X_1 \sim N(\mu,\sigma^2) $ is the same as $\mathbb{E}[X_2^4]$ of $X_2 \sim N(0,\sigma^2)$? If so is there an easy way to show this? I used ...
-1
votes
1answer
26 views

Exercise on bivariate normal distribution [closed]

Let X and Y be normal random variables with mean 0, variance $\sigma^2$ and correlation coefficient $\rho \in (-1,1)$, so that the density is given by $$f(x,y) = \cfrac{1}{2\pi \sigma^2\sqrt{a-\rho^...
1
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0answers
38 views

What is the distribution of $\frac{\sum(x_t-a_t)^2}{\sum(x_t-b_t)^2}$

Let $x_t, t \geq 1$, be a sequence of independent random variables, $x_t \sim N(a_t,\sigma^2), t \geq 1$, $a_t, b_t \in \mathbb{R}$. What is the distribution of $S_n$, where: $$S_n=\frac{\sum_{t=1}^n(...
2
votes
0answers
38 views

Proving two Gaussian random variables are independent given the third: a necessary and sufficient conditon for inverse of covariance matrix

In my probability class I was given this problem that truly has me stumped: Let $ X=(X_1,X_2,X_3) $ be a Gaussian random vector with mean vector zeros, with the 3x3 co variance matrix: $ \...
0
votes
1answer
10 views

$\chi^2$ distribution straight from a joint distribution?

If $(X,Y)$ are normally distributed with mean $0$ and variance matrix $\Sigma$, then why is $$(X,Y)\Sigma^{-1}(X,Y)^T$$ a $\chi^2$ distribution with $2$ degrees of freedom?
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votes
1answer
17 views

Why is probability of infinite sequence of independent Gaussian r.v. in a sphere equals zero for any radius?

Suppose we have a sequence of independent standard normal r.v. $(X_i)_{i=1}^n$. Also, let $S_r$ be an $n$-dimensional ball of radius $r$ centered at zero. Then why does \begin{align} \lim_{n \to \...
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0answers
32 views

Proof of normal distribution property used in Levy's construction of the brownian motion?

I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ...
0
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0answers
3 views

Z-normalization equation - finding the vector before being applied by Z-normalization

I was trying to find a value $X$ that were used to evaluate $Z$ in this equation $$ Z = \frac{X-\mu_x}{\sigma_x} $$ where $X = \{x_1, x_2,..., x_N\}$ and $Z = \{ y_1, y_2,..., y_N\}$ and $$\sigma_x =...
0
votes
0answers
8 views

Distribution of joint from marginals versus joint distribution from marginals

If somebody gives me a variance matrix and a mean vector for a normal distribution, I can read of the marginal distributions. But what if somebody gives me means, and variances, and correlations for ...
0
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0answers
12 views

Existence of asymptotic variance for an estimator when it doesn't converge to normal.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,...)$, where: $U_i=U_i(X_1,...,X_i)$, if for a sequence of constants $\{k_n\}$: $$k_n(U_n-\...
0
votes
2answers
41 views

Evaluation of a tricky integral involving the pdf of a normal distribution

I tried to evaluate the following integral: $$-\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-aw}e^{-\frac{(w-\mu)^2}{2\sigma^2}}dw.$$ It seems that integration by parts does not work. Any help will ...
1
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0answers
97 views

Inequality involving the sum of normal random variables

Problem: Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables from the normal distribution with mean equal to 1.5 and standard deviation equal to 4. Show that with ...
1
vote
1answer
21 views

Maximum Likelihood Estimation of Population Mean(General Normal Distribution)

I am having trouble finding the estimation for the population mean of a generalized normal distribution using maximum likelihood estimation. Using the definition I have: $$l \left(\mu,\sigma^2;x_1,\...
5
votes
1answer
120 views

Computing the inverse error function

I need a formula/series to compute the inverse error function, which is the inverse of $$ \operatorname{erf}(x) = \dfrac{2}{\sqrt{\pi}} \int\limits_{0}^{x} \mathrm{e}^{-t^2} \,\mathrm{d}t. $$ ...
1
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0answers
48 views

Bivariate normal distribution hazard rate

Suppose $(X,Y)$ is bivariate normal with $\mu=\begin{pmatrix} 0 \\ 0 \end{pmatrix}$, $\Sigma=\begin{pmatrix} \sigma^2 & \rho \sigma^2 \\ \rho \sigma^2 & \sigma^2 \end{pmatrix} $ Is it ...
1
vote
1answer
34 views

Distribution function of the quotient of a normal with sum of squares of normals

Let $X$ and $Y$ be independent standard normal random variables and $p\in (0,1)$. Show that $\mathbf P(\frac{Y^2}{X^2+Y^2}\leq p)=2/\pi\arcsin\sqrt{p}$. How to prove this? Apparently the random ...
0
votes
0answers
21 views

I Need Some Help On This Problem Concerning With The Normal Distribution

The Problem: Assume $\mu$ is known. Show this is an exponential family with parameter $\sigma$. The Attempt: If I can show that the function $f(x| \sigma)$ can be written as $a(\sigma)t(x) +b(\sigma)...
0
votes
1answer
28 views

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 ...
0
votes
2answers
95 views

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 ...
1
vote
1answer
17 views

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 ...
2
votes
1answer
32 views

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 ...
1
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0answers
18 views

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 ...
0
votes
1answer
22 views

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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0answers
16 views

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-...
1
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1answer
48 views

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'...
0
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2answers
33 views

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 ...
1
vote
1answer
37 views

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 ...
1
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1answer
27 views

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 ...
1
vote
0answers
41 views

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'...
2
votes
2answers
63 views

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'...
0
votes
0answers
38 views

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 ...
0
votes
0answers
22 views

Transformation of continuous random variables

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 $...
0
votes
1answer
25 views

Verification and Wording

I just need to know if my thinking is correct aka I'm doing it correctly: sometimes the wording trips me up and I second guess myself. The following is the question/summary of it: It's about the ...