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

learn more… | top users | synonyms

2
votes
1answer
90 views

$\chi^2$ test and sampling variance

Let $f(x)$ denote the pdf of a $\chi^2$-distribution with $n\in\mathbb{N}$ degrees of freedom given by $$f(x) = \frac{2^{-n/2}}{\Gamma(n/2)}\cdot x^{n/2-1}\cdot\mathrm ...
0
votes
1answer
94 views

Normal Distribution Transformation

Suppose we have a normal distribution like $ f(x) = \mathcal{N}(\mu = 30, \sigma^2=10) $ and we transform it to another function by multiplying it to $ g(x) = 2x^2 $ the result would be: $ ...
2
votes
1answer
368 views

Closed Form of Normal Distribution

What does closed form in following sentence mean and why we need tables of c.d.f.? Normal distributions's p.d.f. cannot be integrated in closed form, and hence tables of the c.d.f. or computer ...
1
vote
0answers
39 views

Sample estimated normal distribution - what will be the expected effect of another sample?

Assume I already have n samples of a 2D variable. I can compute the sample mean and variance. If I assume that the samples are taken from a normal distribution, then using the mean and variance I get ...
2
votes
1answer
365 views

The distribution of uniformly random rotation of a i.i.d. Gaussian vector $\mathbf{x}$ given $\mathbf{x}$

Suppose that I have vector $\mathbf{x}$ that contains $n$ independently and identically distributed (i.i.d.) zero-mean Gaussian random variables $x_i\sim\mathcal{N}(0,\sigma^2)$. Also suppose I have ...
1
vote
1answer
561 views

prove a bivariate normal distribution

$X$ has a normal distribution. The conditional distribution of another random variable $Y$ given that $X=x$ is a normal distribution with mean $ax+b$ and variance $t^2$, where $a$, $b$, and $t^2$ are ...
0
votes
1answer
42 views

Normal distribution of the dosage

I am having an issue with the following task: For the dose of the MemPro medicine it is known that has normal distribution. Sample of 30 parameters has been taken into consideration and based on the ...
2
votes
1answer
218 views

Conditions for two normal R.Vs to satisfy bivariate normal distribution

I have to Normally distributed Random Variables X and Y which are correlated. What conditions should they satisfy so that their joint distribution is a bivariate Normal distribution?
0
votes
1answer
106 views

Confusion regarding autoregressive process

I was reading this article related to autoregressive processes of order $1$. According to wiki it is given by $$ x_t = \phi{x_{t-1}} + \epsilon \\ |\phi| < 1 \\ x_t|x_1,\ldots,x_{t-1} \sim ...
0
votes
1answer
86 views

$\alpha$-stable distributions and Gauss distribution tails differences.

I know that in a $\alpha$-stable distribution we have: $$ \lim_{x\rightarrow +\infty}f(x,\alpha,\beta)\sim -\alpha \gamma^\alpha \frac{\Gamma(\alpha)}{\pi}sin(\frac{\pi ...
1
vote
1answer
174 views

A simple question about normal distribution

Suppost we have a dataset as below: (Value,Frequency) pairs: (1,2), (2,4), (3,6), (4,8), (5,10) Can we say that this data is normally distributed, or have a normal distribution for this dataset?
1
vote
1answer
320 views

Combination of a normal r.v. with a log-normal one

It is well-known that a sum of normal r.v.'s is another normal r.v., and a sum of log-normal r.v.'s can be accurately approximated with a log-normal r.v. But what can we say if we have a mixture of ...
4
votes
1answer
4k views

Product of Two Multivariate Gaussians Distributions

Given two multi-variate gaussians distrubtions, given by mean & covariance, G1(m1,sigma1) & G2(m2,sigma2), what are the formulae to find the product i.e G1 * G2 ? And if one was looking to ...
3
votes
2answers
737 views

Examples of Student's T-distribution in real world empirical data?

I have recently stumbled onto some empirical (forecasting error) data that should be normally distributed. However, the normal distribution fits relatively poorly due to the abundance of data points ...
1
vote
1answer
96 views

Deriving the characteristic function for $N(0,2)$

Could someone please help me with an easy derivation of the characteristic function for a $N(0,2)$ distribution? Or a link to somewhere it is done.
1
vote
0answers
167 views

Combining general 1D normal distributions into a 2D distribution

My question is a generalization of the question asked here There is a point in 2-D space. I can measure the range of this point from two other locations. I get this measurement as a mean (range) and ...
4
votes
1answer
289 views

Does the integral of PDF of multi-normal distribution over quarter planes have a closed form?

I am interested in finding a closed form solution (wich I suspect does not exist) to the following integral $$\displaystyle \int _a^{\infty }\int _b^{\infty } \frac{\exp \left(-\frac{x^2+y^2-2 c x ...
2
votes
1answer
490 views

“Flattening” a 2D Normal Distribution

I would like to model the probability of a point being at a certain place on a 2D grid. The X coordinate of the point varies according to a normal distribution with mean $0$ and standard deviation ...
1
vote
1answer
668 views

Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent)

Would like to know how to approach this question: Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent). Seen the solution for the ...
1
vote
1answer
576 views

Constructing correlated random variables

How do I construct two correlated random variables with correlation $\rho$ given two i.i.d normal r.v.? Do I just multiply the correlation matrix by a vector generated with two i.i.d normal variables? ...
5
votes
1answer
206 views

Fractional Part of Sum of Sequence of Independent Normal Random Variables

I'm trying to prove that if $X_n$ iid normal $S_n = \sum_1^n X_i$ $U_n=S_n-\lfloor S_n\rfloor$ then $U_n$ is asymptotically uniform in distribution. I've got no idea how to approach this, and it's ...
0
votes
1answer
264 views

Limit of Sum of Cauchy Random Variables

I'm investigating the behaviour of some random variables obtained from standard Cauchy random variables $X_n$. Suppose $Y_n=\textrm{sgn}(X_n)|X_n|^{\alpha}$ for $\alpha\in[0,1]$. Let ...
1
vote
0answers
57 views

A correction to confidence interval.

I have set of random values with the same distribution $y_1, \ldots, y_N$ , $N = mN_1$. $ m \ge 4$, $N_1$ is big enougth( $\approx 1000$ ). I want to to estimeat $E(x)$. How I do it: I make $m$ ...
1
vote
0answers
165 views

Composition of multi complex gaussian normal distribution

assume $w_0$, $w_1$, $w_2$, $w_3$ are circular symmetric complex Gaussian distributions, and the composite of $$ h = e^{j\theta_0}w_0 + e^{j\theta_3}w_3 - e^{j\theta_1}w_1 -e^{j\theta_2}w_2 $$ so ...
1
vote
0answers
211 views

Generating spatially correlated samples from a multivariate normal distribution

I am trying to generate some spatially correlated samples from a multivariate normal distribution following this algorithm Compute Cholesky factorization Q=LL' Sample z~N(0,I) Solve L'v=z Computer ...
0
votes
0answers
244 views

Gaussian difference distribution

let $f_1(x,y)$ and $f_2(x,y)$ be two 2-dimensional Gaussian functions with means $(\mu_{1_x},\mu_{1_y}$} & $(\mu_{2_x},\mu_{2_y}$} and variances $(\sigma_{1_x},\sigma_{1_y}$} & ...
2
votes
1answer
42 views

Is this a place to use Variance, if so what is the meaning of the value?

I want to know if this is a good place to calculate variance in my data,and how to interpret or explain the units of the variance answer. I have 2 lists of corresponding data. ListA has a starting ...
2
votes
3answers
359 views

Find the probability with normal distribution

I am doing some revision for an exam I have tomorrow and I can't work out how an answer is achieved. The weight $X$ kg of a bag of cement can be modelled by a normal distribution with mean $50$ ...
2
votes
0answers
113 views

distribution of block occurrence of random vector in $\mathbb{Z}_2^n$

Given natural numbers $m, n \geq 2$ and a random vector $\mathbf{r}= (a_1,a_2,\cdots,a_n)\in\mathbb{Z}_2^n$. We define the $m$-circulant of $\mathbf{r}$ by the vector ...
4
votes
2answers
3k views

Distribution of the maximum of a multivariate normal random variable

Suppose there is a vector of jointly normally distributed random variables $X \sim \mathcal{N}(\mu_X, \Sigma_X)$. What is the distribution of the maximum among them? In other words, I am interested in ...
8
votes
3answers
13k views

How to calculate the integral in normal distribution?

The factory is making products with this normal distribution: $\mathcal{N}(0, 25)$. What should be the maximum error accepted with the probability of 0.90? [Result is 8.225 millimetre] How will I ...
2
votes
3answers
1k views

$3\sigma$ rule for multivariate normal distribution

I was wondering if the $3\sigma$ rule that holds for 1D normal distribution also holds for multivariate normal distribution?
3
votes
3answers
1k views

$X$ standard normal distribution, $E[X^k]=?$

I'm stuck with a homework problem where we are supposed to prove that the expected value $E[X^k]$, if $X$ has standard normal distribution, is equal to: $$E[X^{2k}]=\frac{(2k)!}{k!\cdot2^k}.$$ But I ...
1
vote
0answers
491 views

Distribution of a squared norm of related multivariate normal distribution.

For $i=1,2,\cdots,2^m$, let $v_i$ be dependent random variables. Suppose for $n$ large, the vector $\mathbf{Z}_n=\left(Z_1^{(n)},\cdots,Z_{2^m}^{(n)}\right)$ with ...
3
votes
1answer
7k views

Why doesn't NORMSINV(RAND()) in Excel work as a standard normal random number generator?

I am looking for an easy way to generate random numbers from a standard normal distribution in Excel. I realize the best way is probably the Box–Muller method, ...
3
votes
2answers
491 views

P.D.F. of independent/dependent Uniform R.V.'s

I am trying to solve this: Consider a stick of length 1. You break the stick in two random places, X and Y. a. Define the individual probability distribution functions of the breaking ...
2
votes
1answer
374 views

CDF of standardized normal R.V.

I'm attemtping to solve this problem: Suppose a shot is fired at a circular target. The vertical and the horizontal coordinates of the point of impact (taking the center of the target as ...
1
vote
2answers
137 views

Independent, Normally Distributed R.V.

Working on this: A shot is fired at a circular target. The vertical and the horizontal coordinates of the point of impact (with the origin sitting at the target’s center) are independent and ...
1
vote
1answer
250 views

Normal Random Variables: Plotting CDF/Variance

So I'm trying to get an edge on Discrete Math for next fall, and I'm working on this problem: Suppose that X is a normal random variable with mean 3. If P{X > 4} = 0.1, what is the variance of ...
1
vote
1answer
328 views

Normal Random Variable - uniform distribution

So here's the question I'm trying to solve: A stock price movement model supposes that if the current stock price is s, then, after one period, the stock price will be $us$ with probability ...
0
votes
1answer
388 views

Evenly distributed normal distribution on surface of a unit sphere

Hello I am using Mathematica code to to generate NPoints of randomly generated points normally distributed around one point on the surface of a sphere using Return[Table[ ...
2
votes
2answers
2k views

Finding percentile rank without knowledge of distribution

Say you take a test and are told your individual score, as a percentage. Eg: 95%. You are also told the aggregate score (average) of all students together, as well as how many students in total took ...
1
vote
0answers
225 views

normal random variable distribution

i have such problem in the book of Applied statistic and probability for Enigneering and need some help to solve it.problem is following: Let random variable X denote a measurement from a ...
0
votes
2answers
1k views

How to integrate $\int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx$?

How can I integrate, $$ \int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx $$ and what's the result w.r.t the Gaussian function's p.d.f $p(x)$ and c.d.f $\phi(x)$? Thanks!
0
votes
3answers
456 views

What does it mean for two random variables to have bivariate normal distribution?

The following is Sheldon Ross's definition: We say that the random variables $X,Y$ have a bivariate normal distribution if, for some constants $\mu_x,\mu_y,\sigma_x>0,\sigma_y>0, ...
1
vote
1answer
506 views

Conditional probability distribution with Gaussian noise

If I have a relationship as follows: $$Y = a X + G(0,\sigma^2),\text{ so }y = a X + \text{some Gaussian noise}.$$ The conditional probability distribution of $y$ given $x$, i.e. $P(y|x)$, is equal ...
1
vote
1answer
159 views

Standard normal distribution probabilities

Ok so I am having difficulty understand the concept behind standard normal distribution probabilities, in the questions I am getting a graph and a table FILLED with numbers, top header column has ...
0
votes
1answer
1k views

How to apply Central Limit Theorem to Uniform Distribution to generate Normal Distrubution?

Suppose I have a simple uniform continuous "unit" distribution X: $$\begin{align*} \forall y \in \mathbb{R} \implies \\ y < 0 : & P(X < y) = 0 \\ y \in [0,1] : & P(X < y) = y \\ ...
0
votes
1answer
212 views

Upper-bound on the sum of two dependent Gaussians.

Let $X$ and $Y$ be two dependent normally distributed continuous random variables (their marginals are $\mathcal{N}(0, 1)$). I would like to find an upper bound on the probability that one is greater ...
1
vote
1answer
371 views

Average Value of Bounded Normal Distribution

Suppose a truck has a capacity of 100 and order sizes to be filled are normal distributed with mean 95 and standard deviation of 10. There is about 30% chance that capacity is exceeded. In this case ...