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

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6
votes
3answers
167 views

Are there order statistics for a Gaussian variable raised to a power?

Let $X$ be a random variable with a standard normal distribution. Let $Y = |X|^{2p}$. I am trying to find the distribution for $Y_{(n)}$, i.e., the largest value of $Y$ out of $n$ samples. I have ...
2
votes
1answer
41 views

Explaining the standard deviation formula

I'm revisiting standard deviation for the first time years, and i can't for the life of me recall the difference between two formulas. In particular, im also looking for how we arrived at these ...
2
votes
1answer
31 views

Calculating transformation of normal random variables.

Let's say you have 4 i.i.d $N(0, 1)$ random variables $X_1 ,X_2, X_3, X_4$, how would you compute the pdf of $\frac{X_1}{\sqrt{X_1^2 + X_2^2 + X_3^2 + X_4^2}}$. I am also interested in the general ...
0
votes
1answer
36 views

Finding the normal distribution using excel

Let's say the lifetimes of a set of tires is normally distributed with a mean of 65,000km and a standard deviation of 6,500km. If a random sample of 9 of the new type of tire are tested, what is the ...
2
votes
1answer
83 views

Tail bounds for maximum of sub-Gaussian random variables

I have a question similar to this one, but am considering sub-Guassian random variables instead of Gaussian. Let $X_1,\ldots,X_n$ be centered $1$-sub-Gaussian random variables (i.e. $\mathbb{E} e^{\...
0
votes
0answers
23 views

Converage rate of product

There are two vectors $\mathbf{a}$ and $\mathbf{b}$ of length $M$ where their entries follow the normal distribution, i.e., $a_i \sim \mathcal{N}(0, \alpha)$ and $b_i \sim \mathcal{N}(0, \beta)$. With ...
1
vote
2answers
63 views

chi squared distribution of independent normal distributions that are not standard normal

I've been working on the following problem. I'm a bit confused about some of the specifics of how to arrive at the correct answer. I hope someone here could point me in the right direction: A dart ...
0
votes
1answer
28 views

Expansion of the Multivariate Gaussian Distribution formula

I'm reading through these notes that claim that if we expand the log of the multivariate gaussian distribution we get the following: $$log(N(x_n|\mu_k, \Sigma_k)) = -\frac{1}{2} (x_n - \mu_k)^T \...
2
votes
1answer
32 views

Calculate $\int_{-\infty}^{\infty}ye^{-\frac{y^2}{2}+xy}dy$

How to calculate $\int_{-\infty}^{\infty}ye^{-\frac{y^2}{2}+xy}dy$ ? (x and y are both real) We know $\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx=1$, but how to apply this ...
-1
votes
2answers
81 views

Let $X$ be a standard normal random variable and $Y = X^2$. [closed]

Let $X$ be a standard normal random variable and $Y = X^2$. Prove that $E[Y|X] = k$, where $k$ is a constant, implies $Cov(X,Y) = 0$. Thanks
0
votes
1answer
42 views

How can the poisson distribution be approximated by the normal distribution?

I know the normal distribution will have both parameters equal to mui, the parameter of the poisson distribution. But why? A vigorous and a less vigorous proof will both help.
3
votes
1answer
56 views

Gaussian process with independent increments

Suppose that we have a continuous Gaussian process $(X_t)_{t \ge 0}$ with independent increments and $X_0=0$. If the increments are also identically distributed, meaning that $X_b-X_a \stackrel{D}{=} ...
0
votes
1answer
44 views

Distribution of the sum of squares of normal random variables

Suppose that $X_1,X_2,...,X_n$ and $Y_1,Y_2,...,Y_n$ are all i.i.d. Normal$(0,\theta ^2)$. What is the distribution of the random variable $T_i=X_i^2+Y_i^2$? And what is the maximum likelihood ...
0
votes
2answers
32 views

Find function for log-normal distribution

I have a set of log-normal distributed values below. I want to find the log-normal curve, which describes the shape of the distribution. I took the default log-normal formular and tried some values (...
1
vote
2answers
28 views

Estimating parameters of a linear model

Suppose there is $n$ data points $(x_i,y_i)$ and $i=1,...,n$, sampled from a line in 2D modelled by $y = m_n x + b_n$ where $m_n \sim \mathcal{N}(0,\sigma^2_m)$ and $b_n \sim \mathcal{N}(0,\sigma^2_b)...
0
votes
0answers
22 views

Measuring normal distribution for weighted data

I want to measure how normally distributed systolic blood pressures are for various large groups of people. There is a problem. Blood pressure values tend to end in 0. If the actual blood pressure is ...
0
votes
1answer
30 views

Projection of a gaussian distribution of dots from a ball to its cross section

I was trying to determine the probability of star collision during a galaxy collision, and came up with this problem : A sphere of radius $R$ and origin $O$ (the Galaxy) has an isotropic Gaussian ...
2
votes
1answer
27 views

What does it mean that the standard normal distribution is invariant under orthogonal transformation?

What does it mean that the standard normal distribution is invariant under orthogonal transformation? This is the context where I found that statement: consider $H\subseteq \mathbb{\mathbb{R}^l}$ a $...
0
votes
2answers
29 views

Expectation of a Standard Normal Random Variable

Calculate E(X^3) and E(X^4) for X~N(0,1). I am having difficulty understanding how to calculate the expectation of those two. I intially would think you just calculate the $\int x^3e^\frac{-x^2}{2} ...
2
votes
1answer
67 views

Is $X^2$ independent from $XY$ where $X$ and $Y$ are standard normals?

I'm thinking they can somehow be expressed as functions of $X-Y$ and $X+Y$, but I haven't quite found out how. Bonus questions: Is it correct that they are both Chi square distributed? And so, ...
0
votes
0answers
23 views

Linear Combination of Multivariate Normal

Given $Y\in\mathbb{R}^N$ follows a multivariate Normal distribution with mean $\mu$ and variance-covariance matrix $\Sigma$. Let $\omega\in\mathbb{R}^N$ be deterministic. I know that $\omega'Y$ is ...
1
vote
1answer
56 views

How to manually create a Z-table

Z-tables are commonly found online. However, I am writing a precision program for this, and so I would like to find out how to calculate my own percentage values.
0
votes
1answer
19 views

Calculate probability of distance for d-dimensional normal

Is there any simple way to calculate the probability of distance in the following form for d-dimensional normal distribution? $P(||\mathbf{x}-\mathbf{\mu}||^2>||\mathbf{x}-\mathbf{a}||^2)$, where $...
-2
votes
1answer
46 views

Scaling property for Brownian motion [closed]

Define Brownian motion as a continuous process $(B_t)$ with independent increments, such that $B_{s+t}-B_{s}$ has normal distribution with mean $0$ and variance $t$. How do you use the independence of ...
2
votes
1answer
44 views

Probability density function and the minimal sufficient statistics for two samples from normal distribution

Suppose $X_1,\ldots, X_m$ is a random sample of size $m$ from the normal distribution $N(\mu_1,\sigma^2)$ with mean $\mu_1$ and standard deviation $\sigma$, and that $Y_1,\ldots, Y_n$ is a random ...
1
vote
2answers
25 views

Find the probability of an event affected by two variables of normal distribution.

A company is producing juice bottles of 2L. One machine is fulling half of the bottle with concentrate juice and an other machine is fulling the other half with water. Let X be the quantity of ...
1
vote
1answer
56 views

Approximate probabilities of a binomial random variable using the normal distribution

A fabrication process is making 8% broken units. Everyday we take 200 units to check how many unites are broken (X). With the normal law as an approximation of the binomial law, find : $$ P (X \le 16)$...
2
votes
1answer
37 views

Product of (dependant) gaussian distributions

I need to find the probability of sampling a specific point on a Gaussian Distribution. The catch is that the mean of the first Gaussian Distribution is itself sampled from a Gaussian Distribution. ...
0
votes
1answer
24 views

Marginilizing in multivariate Statistical Distributions

Suppose we draw $X = [x_1,...,x_i,...x_r,...,x_t...,x_j,...x_N]$ from $N(0,\Sigma)$. is there any way to compute $E[x_ix_j]$ and how about $E[x_ix_jx_rx_t]$ ? How can I compute the marginal ...
2
votes
0answers
35 views

Joint Density and Covariance between Two Random Variables with the same Mean and Variance

This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this. Q1) Are there any general results / relationships to get the ...
2
votes
0answers
65 views

Simple question on conditional probabilities (multidimensional normal distributions)

Let $X$ and $Y$ in $\Bbb{R}^n$ be two random vectors. We assume that $X\mid Y\sim\mathcal{N}(Y,\Sigma_X)$ and $Y\sim\mathcal{N}(\mu_Y,\Sigma_Y)$ The goal is to sample from the distribution of $X$. ...
0
votes
0answers
16 views

Length of Multinormal Distribution

Let $X \in R^n$ be a random normally distributed vector with $E X = \mu$ and $\text{Var } X = \Sigma$. Is there a neat way to calculate the variation of $|X|$, the length of $X$? I know how to ...
0
votes
0answers
16 views

Adding Sum of Squares to Exponent of Multivariate Guassian Distribution

Assume that ${\bf x}$ is an $n\times 1$ vector with a multivariate Gaussian distribution, i.e., for known $\mu_{n\times 1}$ and $\Sigma_{n\times n}$: $$p({\bf x}|\mu,\Sigma) = \mathcal{N}({\bf x}|\mu,...
0
votes
0answers
21 views

Finding boundary point for normal distribution

Suppose that $P(x \, | \, \omega_i) \sim \mathcal{N}(\mu_i, \sigma^2)$ for $i = 1, 2$ and that $P(\omega_1) = P(\omega_2) = 1/2$. I'm trying to find a value $x$ such that $P(\omega_1 \, | \, x) = P(\...
0
votes
0answers
14 views

Singular vectors of random Gaussian matrix

Let $A$ be a singular vector matrix of a random Gaussian matrix. The entries of the Gaussian matrix are i.i.d., so the singular vectors are distributed isotropically. Is it possible to get $E[AA^{H}]$?...
1
vote
1answer
21 views

Using two linear functions of a 3D random vector to find a plane in which it is concentrated

Let us take three random normal variables and combine them into one which we call $X$. We know their means, variances, and covariances, and thus we can come up with a mean vector and a variance matrix:...
0
votes
1answer
26 views

How to find the subspace on which a multivariate normal distribution is concentrated?

Let $(X_1,X_2,X_3)^T$ be a multivariate normal distribution which is singular (that is, its variance matrix has determinant $0$). How can we find the subspace $U \subset \mathbb{R}^3$ with dimension ...
0
votes
1answer
25 views

Question about three Normal distributions

Let there be 3 normal distributions $X\sim\mathcal{N}(\mu_x,\sigma_x)$, $Y\sim\mathcal{N}(\mu_y,\sigma_y)$, $Z\sim\mathcal{N}(\mu_z,\sigma_z)$ and 3 random samples from each distribution- $x,y,z$. ...
0
votes
1answer
43 views

Is $E(\sin Z)=\sin E(Z)$ for $Z\sim N(0,1)$?

Let $Z$ be a random variable of the standarized normal distribution. True or false? $E(\sin Z)=\sin (E(Z))$ ($E$ stands for the mean value). Comments. Since $\displaystyle \sin Z=\sum_{n=0}^{\infty}(...
4
votes
2answers
223 views

Probability of Normal Distribution with Unknown Mean

I am still quite new to the whole idea of probability and statistics and am not sure how to do this question. The random variable R, also normally distributed, has a standard deviation of 3.59 with ...
1
vote
1answer
23 views

Integrating inverse cumulative of standard Normal Distribution

I am studying this book and a particular line reads $\int_{1-p}^1 \Phi^{-1}(u)du=${set $k=\Phi^{-1}(u)$}=$\int_{\Phi^{-1}(1-p)}^\infty k\phi(k)du$, where $\Phi$ and $\phi$ are the cumulative and ...
1
vote
1answer
35 views

creating positive definite matrix using wishrnd in matlab

I used the below code to create a matrix of wishart distribution, ...
0
votes
1answer
15 views

Modelling a Bivariate Normal Distribution in Matlab

Hi I was wondering would anyone have experience with modelling multiple bi-variate Gaussian distributions on the same plane in Matlab? Say we know the coordinates of the means, let m1 = mean1= [1 2] ...
0
votes
0answers
25 views

How to calculate sample variance?

I have an infinite population. Out of this population, I'm picking $N=40$ items, each of which can have a value of $0$ or $1$. Say I get a mean of $0.7$. How do I calculate the standard error of this ...
0
votes
1answer
25 views

Recover Marginal Distribution subject to a Constraint

I want to identify the marginal of a normal distribution subject to a restriction. Take two normally distributed random variables $x,y$. Their pdfs are denoted by$\phi(x)$ and $\phi(y)$. The moments ...
2
votes
1answer
39 views

Exponential law with both positive and negative values

The exponential law with density $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ and $f(x)=0$ for $x < 0$, is well-known. What's the name of the distribution which has $$f(x) = \frac{1}{2} e^{-|...
1
vote
1answer
25 views

on a quantisation of the bell curve

The bell curve function: $e^{-x^2/2}$ is an eigenfunction of the Fourier transform (FT) on the real line. Is its quantisation/discretisation the binomial distribution (coefficients $n$ choose $k$) an ...
1
vote
1answer
78 views

Multivariate normal value standardization

I am wonder how to standardize multivariate normal value. Normal standard multivariate distribution of $q$ variables is $z\sim N_q(0_q,I_q)$. I have found that $Bx\sim N_q(Ba,B\Sigma B^T)$ and based ...
2
votes
1answer
75 views

Maximizing the probability of a poll prediction

Using the central limit theorem, I was able to find out the first part of this question. However, part b is eluding me. How do I, in general, find a value for $n$ such that we can ensure the ...