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

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3
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0answers
237 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
0
votes
1answer
28 views

“Distance” of iid gaussian variables [duplicate]

Take two i.i.d. Gaussian R.V.s $X$ are $Y$ both of which are $~N(0,a\sigma)$. Define a new R.V. $D = \sqrt{X^2 + Y^2}$. What's the expected value $E(D)$? In researching this I'm seeing references ...
0
votes
1answer
37 views

Variance of not quite the product of two independent, normally distributed random variables

Let's say I have two independent variables, $X\sim N(10,9)$ and $Y\sim N(5,4)$. $X$ represents the number of orders received in a month, and $Y$ represents the size of each order. For this example, a ...
1
vote
1answer
343 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this $$...
0
votes
1answer
359 views

Product of two multivariate Gaussian pdfs - normalizing constant

https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf contains expressions (p.2, 6e, 6f) for the normalization constant for the product of two multivariate Gaussian pdfs, with mean vectors $a$ and $b$ ...
0
votes
1answer
250 views

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: ...
0
votes
1answer
38 views

How to approximate a normal distribution?

Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the ...
0
votes
1answer
159 views

Variance with minimal MSE in normal distribution

Given $X_1,...,X_n$ ~ i.i.d. $N(\mu, \sigma^2)$ where the mean is unknown, let the estimator for $\sigma^2$ be $\hat{e} = p\sum_{i=1}^n(X_i-\overline{X})^2$ How do I choose $p$ so that this estimator ...
0
votes
0answers
31 views

First order moment of multivariate Gaussian random vector

Let $X = (X_1,\dotsc, X_n)$ be a random vector distributed as a multivariate Gaussian with mean $0$ and covariance $\Sigma$. What is $\mathbb{E}[X_1\dots X_n]$?
0
votes
2answers
60 views

Covariance between $X$ and $Y$ of a bivariate normal distribution?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X$= 5 mL, $\sigma_Y$= 2 mL, $\mu_X$= 120 mL, $\mu_Y$= 100 mL, and $\rho$ = 0.6. How do I find the covariance of $X$ and $Y$? I know the ...
1
vote
1answer
36 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
1
vote
0answers
33 views

If B is a N(0,1) R.V., show $E[B^4] = 3$

I've read in Elementary Stochastic Processes by Mikosch (p. 98), that it is a well known fact that: If B is a N(0,1) R.V., $E[B^4] = 3$ I also see something equivalent (but uncited) on the ...
0
votes
0answers
301 views

Expected Value of the absolute value of the sum of random variables

Hi everyone and thanks in advance. Let's say we have a random variable Y which can be expressed as the sum of two other complex random variables X and W, i.e. $ Y = X + W $. $X$ and $W$ are ...
1
vote
0answers
24 views

Scaled distribution of Brownian motion

If I have $X = 5(B_t - B_s)$ Does this have a distribution of $\sim \text{N}(0,25(t-s))$ ? Since $B_t - B_s$ has distribution $\sim \text{N}(0,t-s)$ Then $X = \mu \cdot 0 + \sigma_1 Z$ where $Z \...
0
votes
1answer
43 views

Frequency Distribution and Throughput

I am conducting an experiment on a couple of computer systems but the results I have don't make sense to me. I made each system perform 1000 operations: System A performs operations at a rate of <...
0
votes
1answer
40 views

The product of multiple univariate Gaussians

What is the final result of $$I=\mathcal{N}_{x}(\mu_1,v_1)\,\mathcal{N}_{x}(\mu_2,v_2)\ldots\,\mathcal{N}_{x}(\mu_n,v_n)=\frac{1}{\sqrt{2\pi\,v_1} } e^{ -\frac{(x-\mu_1)^2}{2v_1} } \frac{1}{\sqrt{2\pi\...
1
vote
1answer
34 views

How to extract a covariance matrix with this information

Referring to the above image, I wanted to know how to get the covariance matrix $\sum$. My understanding is, $A$, is our transformation matrice, such that $\begin{bmatrix} X_1 \\ X_2 \\ \end{bmatrix}...
0
votes
1answer
67 views

Product of two densities, when one of them is “incomplete”

One can frequently read that the product of two multivariate Gaussian pdfs, $f_1(x)$*$f_2(x)$, is itself a Gaussian function, with parameters as defined for example in: http://www2.imm.dtu.dk/pubdb/...
1
vote
1answer
23 views

Linear transforms of Normal dist [closed]

If $X_t = \sqrt{t} Z$ where $Z \sim \text{N}(0,1)$ Then show the distribution of $X_t - X_s$ for $s<t$ Just wanted to check, would this be $\sim \text{N}(0,t-s)$ or $\sim \text{N}(0,(t-s)^2)$ ?
0
votes
2answers
42 views

Distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion

I am trying to find the distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ...
0
votes
1answer
35 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...
0
votes
1answer
48 views

Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
1
vote
1answer
58 views

distribution of distance between two points whose coordinates are normal random variables

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?
0
votes
1answer
96 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
0
votes
2answers
69 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
0
votes
1answer
147 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...
0
votes
1answer
88 views

Calculate P-Value

In a certain area, regulations require that the chlorine level in wastewater discharges be less than 100 $\mu$/L. In a sample of 85 wastewater specimens, the mean chlorine concentration was 98 $\mu$g/...
0
votes
2answers
185 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
0
votes
1answer
71 views

Expectation of a linear combinations of iid standard normal, restricted to a halfspace

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there any easy way to calculate $$\mathbb{E} \left[ 1_{\displaystyle \left\{ \sum_{...
2
votes
0answers
90 views

How to fit normal cumulative distribution functions

For a normal distribution $N(\mu,\sigma^2)$, we know its cumulative distribution function is $F(x)=\Phi(\frac{x-\mu}{\sigma})$ where $\Phi(x)$ is $cdf$ for standard normal distribution which means $$ ...
1
vote
1answer
60 views

Calculating probabilities for complex random variables

I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: $Y=10(2+\mu+\...
0
votes
0answers
416 views

The X & Y coordinates for points on a bell curve / normal distribution?

In Short: I want to give a formula the X coordinate and get the Y coordinate from matching a bell curve. Is this possible? In Detail: I'm trying to program a market simulation and to get a product's "...
2
votes
2answers
53 views

How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
4
votes
1answer
117 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let $...
-1
votes
1answer
356 views

Normal distribution with dice

I'm wondering how to control the normal distribution that comes from summing dice rolls only using different numbers of dice, different combination of types of dice (d4, d6, d8, d10, d12, d20) and ...
2
votes
1answer
125 views

Expectation of product of two correlated gaussian variables

$\newcommand{\var}{\operatorname{var}}$It seems I can not find the answer anywhere, please point it out how to calculate. Here, I have $X$, $Y$,$G$,$X_D$ and $Y_D$,both are Gaussian variables, and $...
0
votes
1answer
51 views

Variance of a Gaussian Random Variable

Show Variance of a Gaussian random variable $N(\mu,\sigma^2)$ and I know $\mathbb{E}(X)^2 = \mu^2$. So I need $\mathbb{E}(X^2)$ = $\int_{\mathbb{R}} x^2 \frac{1}{\sqrt{2\pi\sigma^2}} e^\frac{-(x-\mu)...
0
votes
1answer
126 views

Sigmoid function that approaches infinity as x approaches infinity.

The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity. The ...
2
votes
0answers
77 views

Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
1
vote
1answer
31 views

finding variance of gaussian distribution from mean

The Gaussian random variable $X$ can be used to model the number of customers that enter a market in 1 minute at a given time of the day. The mean number of customers that enter the market in 1 minute ...
1
vote
3answers
1k views

Finding the probability of loss from standard deviation in normal distribution

I am unsure how to approach the following question. The returns from a project are normally distributed with a mean of \$220,000 and a standard deviation of \$160,000. If the project loses more than \...
0
votes
1answer
21 views

Representation of a non-standard normal variable squared

I have come across a representation of a non-standard normal distributed variable square. It is clear for me that assuming $Z_j \approx N\left ( \theta_j, \frac{\sigma^2}{n} \right )$ we can write ...
0
votes
1answer
50 views

normal approximation of binomial distribution

a school buys 60% of its light bulbs from supplier A and 40% from supplier B. the light bulbs from both suppliers look identical but light bulbs from supplier A have exponentially distributed ...
1
vote
2answers
162 views

Gaussian distribution raised to a power

Given that $X$ follows a Gaussian distribution $e^{-x^2/2\sigma^2}$, what distribution is followed by $X^{1/3}$? How does one start to solve this problem? I guess it isn't $e^{-x^{2/3}/2\sigma^{2/3}}...
-1
votes
1answer
84 views

How to Normalize the Sum of Two Gaussians

I have the following function: $I(\theta_i) = I_0 + I_1\exp(\mu(\cos(\theta_i - \theta_s) - 1))$. Suppose I have two implementations of this function, whose parameters match with the exception of $\...
2
votes
0answers
35 views

Characteristics function and moments of multivariates

I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf This is a paper by Nengjiu Ju who uses talyor series to express ...
0
votes
1answer
141 views

Calculate the Probability of a Normally Distributed Random Sample

Please i would like to understand these problems about probability distributions, I can't find a right solution for this problem. I have a variable X which is the level of glucose in blood and is ...
0
votes
0answers
105 views

expected value minimum of bivariate normal distribution

Let $X,Y$ be jointly normal with density $f(x,y)=\dfrac{1}{2\pi\sqrt{1-\rho^2}}\exp(-\dfrac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2))$. Let $Z=\min(X,Y)$. Show that $E[Z]=\sqrt{\dfrac{1-\rho}{\pi}}$ and $E[Z^...
1
vote
1answer
24 views

Understanding the normalization of a Gaussian

I have a Gaussian defined as follows: $W(\theta) = j * exp(-0.5 * \theta^2 / \sigma^2)$. I want to set $j$ such that $\frac{1}{360}\int_{-180}^{180} W(\theta)d\theta = 1$. I'm using two values for $...
2
votes
1answer
43 views

variance of multivariate normal

currently trying to compute the first two moments of the multivariate distribution. Got an extremely helpful answer to show that $\mathbb{E}[x]=\mathbb{m}$, with $x \sim \mathcal{N}(\mathbf{m},\mathbf{...