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

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0
votes
2answers
182 views

Conditional multivariate normal pdf with inequality $f(x_1 | x_2 > a)$

Let $$\begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim\mathcal{N}\left[\begin{pmatrix} 0\\ 0 \end{pmatrix} ,\begin{pmatrix} \sigma_{1}^2 & ...
2
votes
1answer
72 views

Is my understanding of the Central Limit Theorem correct?

Have I got this correct - Say we have a population. We take a random sample of size $n$ from this population. I.e. we form a sample $S$ based on random variables $X_1, X_2, ..., X_n$ taken from this ...
0
votes
1answer
184 views

cumulative standard normal distribution formula

I need to calculate a P-value (for significance checking) out of the Z value, mean(0), standarddeviation(1), normal distrubution being cummulative. Is there a function in PHP that could do that? ...
1
vote
1answer
54 views

Probability of random variable in Normal Distribution

I've been talking to my lecturer about choosing random values from a Normal Distribution and he says the following: "Roughly 68% of expected values $ \in (\mu-\sigma,\mu + \sigma)$ does not imply ...
0
votes
1answer
68 views

How Moment generation function of Gaussian R.V. can be divided into this one?

I know the moment generating function of Gaussian random variable is $$E\{e^{rx}\}=\int^{+\infty}_{-\infty}e^{rx}f(x)dx=e^{mr+r^2\sigma^2/2}$$ where $f(x)$ is PDF of Gaussian R.V., $m$ is mean value ...
0
votes
1answer
53 views

variance of a random variable

If $X_1, X_2 , ....., X_n$ iid $N(0,1)$ , and $S^2$ was defined as the population standard deviation we are to find the variance of $S^2$ I want to know the distribution in order to find the ...
0
votes
2answers
413 views

moment generating function technique

If $X$ was a random variable with a distribution $\mathrm{Normal} ( 0, 1 )$, using moment generating function technique we have to show that $Y= X^2$ has the Chi-square distribution with $1$ degree of ...
1
vote
2answers
56 views

Sampling from a Normal Distribution

If I am sampling randomly from only the -sigma to +sigma interval of a normal distribution and rejecting all other numbers, does it imply that the probability density changes? If so, by what degree? ...
0
votes
2answers
45 views

$P(X>16|X>10)$ - normal distribution

If $X$ is a normal random variable with parameters mean = 10 and standard deviation= 6, compute $P(X>16|X>10)$ ? Can someone help to explain this $P(X>16|X>10)$ in the normal rv. term? ...
1
vote
1answer
169 views

Convergence of a sequence of Gaussian random vectors

Let $X_n$ be a sequence of Gaussian random vectors that converges in distribution to some random vector $X$. Is $X$ Gaussian? Partial solution If we can show that $\mu_n := E\left(X_n\right)$ and ...
1
vote
2answers
100 views

Statistics - Lost with this question

I'm having trouble doing this question because I don't know where to begin. Could someone walk me through this slowly so that I understand the thought process and how to approach questions like this? ...
0
votes
1answer
50 views

Expected Value of Exponential

I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
1
vote
1answer
36 views

Upper bound for the gaussian measure of an epsilon strip.

I have a question concerning the normal probability distribution: Suppose that $X\sim N(\mu,\sigma)$ is a normal distributed random variable with mean $\mu$ and variance $\sigma$. Let ...
0
votes
1answer
342 views

Gaussian distribution and its parameters

I need to learn more about Gaussian distribution and given a set of data, plot a Gaussian distribution of it. Using the following code sample, could you please tell me how I can plot a Gaussian ...
1
vote
1answer
2k views

MLE of bivariate normal distributoin

Suppose that $X$ ($n$ by $2$ matrix) follows a bivariate normal distribution $N(\mu,\sigma^2I)$, where $I$ is the $2\times 2$ identity matrix. How to find the maximum likelihood estimates of $\mu$ and ...
1
vote
2answers
295 views

Hypothesis test (when to use z-test or t-test)

I've got 6 questions here. I don't really need the complete answers, I just want to know what test (z or t) should be used and what are the basis for using that test. Here we go: (1) According to the ...
1
vote
2answers
72 views

Use z confidence interval to estimate population proportion

Which of the following must be true of a sample in order for it to be appropriate to use a $z$ confidence interval to estimate the population proportion? (A) The sample is a random sample from the ...
4
votes
2answers
1k views

Convolute exponential with a gaussian

I have data measuring an exponential decay that is convoluted by a gaussian response function. I have the measured shape of the gaussian, and want an analytical expression for the exponential ...
1
vote
1answer
43 views

How to show that the method to create two correlated random numbers is correct?

I would like to understand how I can show that the method to create two normally distributed random numbers given as an answer to this question is correct. Given independent $X_1$ and $X_2$ normally ...
0
votes
2answers
85 views

Maximum of two skewed normal distributions

Does there exist a means to approximate the maximum of two skewed normal distributions in terms of another skewed normal distribution? To make it clearer, given 2 skewed normal distributions ...
3
votes
1answer
299 views

Exercise on Conditional Expectation of Jointly Gaussian Random Variables

I am trying to solve the following exercise from my professor's notes on conditional expectation: Let $x: \Omega \rightarrow \mathbb{R}^n$, $x \in G(0, Q_x)$, $Q_x = Q_x^T>0$, $y: \Omega ...
3
votes
1answer
224 views

Distribution of higher powers than 2 of a gaussian distribution

If $X \sim \mathcal{N}(0,1)$, then $X^2 \sim \chi^2(1)$. What about higher powers of $X$? I know that the Gamma Distribution is a generalization of the $\chi^2$ distribution, but I don't know how the ...
0
votes
1answer
24 views

Median of limited normal distributions

If $$x + \mathcal{N}(0,\,1) = 6$$ $$y + \mathcal{N}(0,\,1) = 7.5$$ $$\left|{y-x}\right| < 1$$ then what is the median of $y$? I'm expecting it to be $< 7.5$. If it is not, then why doesn't the ...
0
votes
1answer
31 views

Bound for the Ratio of Standard Normal to Standard Cauchy

Let a(x) = N(0,1) and let b(x) = C(0,1). I would like to prove that the ratio of $$\frac{a(x)}{b(x)} \le \sqrt{\frac{2\pi}{e}}$$ Doing algebraic simplification yield something like $$(1 + ...
2
votes
4answers
2k views

Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 ...
1
vote
1answer
81 views

A sequence of Gaussian random vectors converges to a Gaussian random vector

Suppose $\left\{X_n : n \in N\right\}$ is a sequence of Gaussian random vectors and $\lim_n X_n = X$, almost surely. If $b := \lim_{n\rightarrow\infty} EX_n$ and $C := \lim_{n\rightarrow\infty} ...
0
votes
1answer
125 views

Find the MLE of bivariate normal

Suppose that $X = (x_{ij})n*2$ follows a bivariate normal distribution $\mathcal{N}(\mu, \sigma^2I)$, where I is the $2\times 2$ identity matrix. How to find the maximum likelihood estimates of $\mu$ ...
1
vote
1answer
493 views

Central Limit Theorem exercise

I'm trying to solve this exercise: Drums labeled 30 L are filled with a solution from a large vat. The amount of solution put into each drum is random with mean 30.01 L and standard deviation 0.1 L. ...
2
votes
2answers
902 views

X follows an exponential distribution, calculate Expected value of sqrt(X).

Problem: Let X follow an exponential distribution with expected value of 1. Define Y=sqrt(X). Calculate E(Y). This is my first course in probability theory (5 weeks ≈ about 5*40 hours of workload) so ...
0
votes
1answer
31 views

Two different ways to calculate the probability for a negative value - are they equivalent?

I want to calculate $$ P(Z\leq-1.8) $$ My math book teaches this one: $$ F_Z(0) - F_1(1.8) = 0.5 - F_1(1.8) $$ This makes sense. But what about the following? Does it also make sense, is it ...
1
vote
0answers
29 views

We said the data is normally distributed, based on the raw data or residual?

I have a confusing regarding the assumption test for the data, in some theory were said that there are three assumption of data as we called as "good" data: Independent Normally distributed ...
0
votes
2answers
2k 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
3answers
116 views

Computing standard deviation of discrete normal distribution

I used below pseudocode to generate a discrete normal distribution over 101 points. ...
0
votes
0answers
249 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: ...
0
votes
0answers
19 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
110 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
36 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
163 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
118 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) ...
1
vote
1answer
127 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
136 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
43 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
152 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
36 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
81 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
54 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
24 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
21 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
156 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 ...
1
vote
0answers
33 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 ...