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

learn more… | top users | synonyms

0
votes
2answers
1k views

Are any linear combination of normal random variables, normally distributed?

It is easy to show that if we have n independent normally distributed random variables, then a linear combination fo them ar normally distributed. It is also said that if (x1,x2,..,xn) is ...
0
votes
0answers
40 views

Folded normal distribution

I am using bayesian stats for image classification and I have a variety of input variables. These inputs are normally distributed except for two. One has a truncated normal distribution and the other ...
0
votes
3answers
80 views

Computing standard deviation of discrete normal distribution

I used below pseudocode to generate a discrete normal distribution over 101 points. ...
0
votes
0answers
124 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
1
vote
0answers
17 views

The space of all normal covariances matrices

Let $\cal C$ be the space of all $k-$variate normal covariance matrices and $\cal M$ be the set of all $k\times k$ symmetric positive semi-definite matrices. As we know that if $k=1$ then ${\cal ...
0
votes
1answer
78 views

Using Chi-Square to test normality.

This is a sample question we received. I can't really figure out how to statistically show that this data is normally distributed. We are to used the chi-square method and these are the steps we are ...
0
votes
1answer
31 views

bhattacharrya distance

I have two bivariate Gaussian distributions with row-vector means and a 2x2 covariance matrix for each. I am trying to find what the following equations are doing, and ultimately type of value it is ...
1
vote
2answers
102 views

Distribution of sum of jointly normal random variables with given covariance matrix

Assume that $(X_1, X_2, X_3)$ are jointly normal random variables with the mean vector $(a,b,c)$ and the covariance matrix: $$\left( \begin{array}{ccc} \sigma_1^2 & \alpha & \beta \\ \alpha ...
0
votes
2answers
114 views

The problem of x = ln(x)

I am trying to find x values for points along the normal distribution curve, and I ended up with a problem that goes back to the method of solving $x = \ln x$. Right now, I have $\ln(a \mu) - \ln(10) ...
0
votes
0answers
26 views

Trying to add travel impact to a predictive least squares model

Second time poster and I'm not 100% sure if I should be posting this in the maths or stats section. Basically I run least squares based models on most sports (just for fun - I'm not great at maths) ...
1
vote
1answer
105 views

help with Borel Cantelli lemma

There is a sequence of random variables $X_1,X_2,...$ For each i $X_i$ ~ $Normal(0,1)$ Is $ \frac{X_n}{n} \rightarrow 0 $ almost surely? Is $ \frac{X_n}{lnn} \rightarrow 0 $ almost ...
1
vote
1answer
87 views

Probability of Cholesterol levels

If the mean serum cholesterol level is 217 and the variance is 750, then what is the probability that a randomly selected person would have: A. Cholesterol value between 150 and 250 B. Greater than ...
0
votes
1answer
38 views

About independence and dependence of normal distributions

I encountered two interesting questions: $A=X+Y, B=X-Y$, where $X$ and $Y$ are independent standard normal distribution. Can I draw the conclusion that $A$ and $B$ are independent because they ...
1
vote
1answer
133 views

Is data normally distributed at the 5% significance level?

I have a statistics question I cant wrap my head around: The data sure looks normally distributed as it follows a bell curve and the mean, median, mode could are relatively the same. I just don't ...
1
vote
0answers
34 views

Frechet differentiability, asymptotic normality

I try to prove the asymptotic normality from the Frechet differentiability. Consider $$T(G)-T(F)=L_{F}(G-F)+o\left(d_{\star}(G,F)\right)$$ and ...
0
votes
1answer
68 views

How can we derive expectation of two dependent normal distribution?

$\mathbf{X}$ and $\mathbf{Y}$ are each dependent normal random variable, then how can we derive like this one? $$\mathbf{E}\{e^{\mathbf{X}}e^{\mathbf{Y}}\}$$ I know the each first moment is ...
0
votes
1answer
38 views

Verify that moments of gaussian variable are given by a formula

I would like to ask you to verify if the following statement is true. Let $X$ be a normal-distributed R.V. with $0$ mean and $\sigma ^2$ variance. Then $$ \mathrm{E}\left[X^p\right] = ...
0
votes
1answer
17 views

statistics - multivariate normal distn, variance and probability of event?

I have a multivariate Normal distribution defined by: μx = 360, μy = 280, μz = 180, σx = 40, σy = 34, σz = 48, and correlations of ρxy = −0.41, ρxz = −0.34, and ρyz = 0.47. I am required to find ...
1
vote
0answers
19 views

Linear Gaussian system, covariance of the normalisation constant

If we have the following multivariate Gaussian distributions: $$p(x) = N(x|\mu_x,\Sigma_x)$$ $$p(y|x) = N(y|Ax + b, \Sigma_y)$$ Now how can you deduce p(y) ? p(y) is called the normalisation ...
3
votes
2answers
126 views

If $X$ is normal, is $\exp(X)$ still normal? How to find its mean and variance?

$X$ is a random variable for normal distribution: $X\sim N(\mu, \sigma^2)$. What is the mean and variance of $\exp\{x\}$? My attempt: $$E[\exp\{x\}]=\exp \{E[x]\} \text{, by the invariance ...
0
votes
0answers
13 views

Two Way Random Effect Model. $\mathbb E(\alpha_i\bar \epsilon_{i..})$?

$\alpha_i$ is random effect of $i$th level of factor $A$ and $\alpha_i\sim NID(0,\sigma_{\alpha}^2)$ $\epsilon_{ijm}$ is the random error term of $i$th level of factor $A$ and $j$th level of factor ...
0
votes
0answers
27 views

intergral of the product of 2 multivariate Gaussian distribution

Suppose there are the following relationships between $x,y,w$, $$\begin{align}p(x,y) &= N(\mu_1, \Sigma_1)\\ p(x\mid w) &= N(\mu_2,\Sigma_2)\end{align},$$ is it possible to compute $p(y\mid ...
1
vote
1answer
46 views

Normal and standard distribution

There is some details i don't understand in my book, here goes; Let $X \sim N(\mu,\sigma^2)$ and $Z\sim N(0,1)$ we know that: $$F_X(x) = \int\limits_{-\infty}^{x} \frac{1}{\sigma ...
3
votes
1answer
88 views

Bound on the $Q$ function related to Chernoff bound

For the function $Q(x) := \mathbb{P}(Z>x)$ where $Z \sim \mathcal{N}(0,1)$ \begin{align} Q(x) = \int_{x}^\infty \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2} \right) \text{d}u, \end{align} for ...
0
votes
2answers
72 views

What is the probability that a Chi-square distribution lies within 2 standard deviation of its mean?

Here I have an 8 degrees of freedom Chi-square distribution function $f(x)$ So by definition, $E(X)=8, Var(X)=2*8=16$. (Please guide me if this is wrong. We just started this chapter and there's ...
1
vote
1answer
3k views

'normally distributed random numbers' vs 'uniformly distributed random number'?

what is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)
1
vote
1answer
1k views

Distribution of the sum of squared independent normal random variables.

The sum of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then ...
2
votes
1answer
301 views

Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
0
votes
1answer
28 views

If each $X_i$ is distributed $N(0, \sigma^2)$, what is the distribution of $\sum_{i=1}^k X_i^2$?

I tried it for $k=1$ with an integral, so: $P(X^2 <x) = \int_{-\sqrt{x}}^{\sqrt{x}} \frac{1}{\sqrt{2 \pi} \sigma} e^{\frac{t^2}{\sigma^2}} dt$, but this didn't work out. I suspect there is faster ...
0
votes
1answer
105 views

how to generate Normally distributed random number?

I am looking for a function that can generate Normally distributed random numbers. I came to know about bux-muller transform but I didn't understood it completely what it is doing. Thus it would be ...
0
votes
0answers
5 views

Generating distribution from clusters

I am working on image processing where I have 15 clusters corresponding to 3 dimensional points. These points are clustered according to the 15 fixed variables over a duration. (for example 10 ...
0
votes
1answer
100 views

Weighted mean from a set of average and standard deviation pairs

I'm trying to replicate some math a professor did related to Twitter sentiment analysis. Basically, there is a sentiment dictionary, called ANEW, that contains a mean and standard deviation for 3 ...
0
votes
0answers
55 views

Finding the joint posterior distribution of AR(2) process

Suppose we have AR(2) process for $\{y_t, t=3,4,..\}$ and let $a_1,a_2,\sigma^2$ be the parameters of the time series. We assume that $y_1$ and $y_2$ er independent normally distributed with mean zero ...
0
votes
1answer
36 views

Finding the z value

So I am new to normal distribution, I need help on understanding on how to calculate it's z-value Question : What is the z value which has 87.49% of the area below it?
12
votes
3answers
215 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
3
votes
0answers
94 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
0
votes
1answer
387 views

Q function and the Chernoff bound

How do we use the Chernoff bound to prove that $$ Q(x)\leq e^{-\frac{x^{2}}{2}} $$ where Q(x) is the probability that a standard normal random variable X takes a value greater than x
0
votes
0answers
19 views

Sampling distribution with large sample size

As the sample size $n$ of a sampling distribution of sample means increases, the distribution becomes more normal. But if $n$ were the same size as the (finite) population, the "sampling" distribution ...
0
votes
2answers
119 views

Adding two normal distribution

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. And Suppose that $Z \sim N(1, 2^2)$ and is independent of all $X_i$. Define $Z_i = Z + X_i$ for $i = 1, ...
0
votes
2answers
60 views

Central Limit Theorem Application on Multivariate Normal

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. What is the distribution of $\overline{X} = \frac{1}{3}(X_1+X_2+X_3)$? I don't quite understand how to ...
1
vote
1answer
346 views

How is the entropy of the multivariate normal distribution with mean 0 calculated?

Here is what I have so far: $$\begin{align} h(x) &= - \int \frac{1}{(2\pi)^{\frac{D}{2}}\det\Sigma^{\frac{1}{2}}} \exp(-\frac{1}{2} x^T\Sigma^{-1}x) \ln ...
2
votes
1answer
118 views

The radial part of a normal distribution

I am reading a paper that asks me to sample $s_i$ from a distribution like this: $s_i \sim (2\pi)^{-\frac{d}{2}}A^{-1}_{d-1}r^{d-1}e^{-\frac{r^2}{2}}$ "Here the normalization constant $A_{d−1}$ ...
2
votes
0answers
109 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
2
votes
1answer
475 views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
0
votes
2answers
62 views

MGF/ expectation Gaussian Random Variables

I am stuck with something that seems easy but i cannot recall how to figure it out? Let $G_1$ and $G_2$ be two standard gaussian random variables with mean $0$ and variance $1$. Then how to calculate ...
0
votes
1answer
131 views

Combining discrete and continuous random variables

Here is the question: $X$ has the distribution $\mathcal{N}(0,1)$ and $Y$ is such that $P(Y=1) = P(Y=-1) = \frac{1}{2}$ Suppose that $X$ and $Y$ are independent and that $Z = XY$. Are $Y$ and ...
0
votes
1answer
50 views

Normal Distribution, cant seem to reach the right answer

The scores on a statistics test are Normally distributed with parameters mean = 80 and standard deviation = 196. Find the probability that a randomly chosen score is no greater than 70 My attempt, ...
0
votes
2answers
61 views

Random variables in normal distribution

Suppose that $X_1$, $X_2$ are independent $\mathcal{N}(0,4)$ random variables. Compute $P\left(X_1^2<36.84-X_2^2\right)$. I have no idea how to start this. Do I have to do anything to the ...
1
vote
2answers
177 views

Distribution of angle of two dimensional normal vector

The original subject is: Suppose random variables $X$ and $Y$ are independent and both follow the Normal distribution $N(0,\sigma ^2)$. 1) Prove $U=X^2+Y^2$ and $V = \frac{X}{\sqrt{X^2+Y^2}}$ ...
1
vote
0answers
57 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...