2
votes
0answers
17 views

Write $\Phi_n(\sqrt{y-1})$ in terms of $\Phi(y)$ and $n$. ($\Phi_n$ CDF of a $\mathcal{N}(0,\frac{1}{n})$)

I'm trying to solve the following problem: Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt $$ Let $F_{Y_n}$ ...
1
vote
2answers
23 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
0
votes
0answers
18 views

p.d.f and distribution of multivariate normally distributed variables

Suppose $X\sim N(\mu,V)$ where $\mu = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$ $V = \begin{pmatrix} 3 & 2 & 1 \\ 2& 4 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ a) ...
3
votes
1answer
127 views

Finding the distribution function of a random variable using CLT

Let $f_0$ and $f_1$ be two continuous probability density functions with means $\mu_0,\mu_1$ and variances $\sigma_0^2,\sigma_1^2$ on $\mathbb{R}$. Furthermore, let $l(y)=f_1(y)/f_0(y)$ be the ...
1
vote
1answer
51 views

Bivariate distribution of the sum and product of Gaussian distributed numbers

If $X$ and $Y$ are independent normally distributed random variables $$X,Y\sim\mathcal{N}(0,\sigma^2)$$ How are the sum and product, $X+Y$ and $XY$, co-distributed? You can write the moment ...
0
votes
0answers
3 views

Finding a Gaussian Distribution to approximate a distribution with non-positive definite covariance matrix

We have got a Gaussian distribution covariance matrix(precision matrix) and the potential information, that is, if g is proportional to exp(-X'KX+h'X). However, K here is not positive definite. So we ...
0
votes
1answer
16 views

Moments of maximum of bivariate standard normal

Let $X,Y \sim N(0,0,1,1,\rho): f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}}$, and let $Z=max\{X,Y\}$. I'm looking for the first two moments of $Z$. I know it is ...
0
votes
1answer
14 views

Bivariate normal distribution when $\rho$ is 0

What happens to the bivariate normal distribution when $\rho$ is 0?The bi-variate normal reduces to a simpler distribution, but what is it? and how do you calculate the cdf then? What I have tried: ...
0
votes
1answer
40 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$
0
votes
0answers
14 views

Parameters of normal distribution following other distributions

x follows a normal distribution: x~Normal(μ,σ). However, the two parameters of this normal distribution, μ, σ, follow other distribution. Specifically, μ follows normal distribution: ...
2
votes
2answers
50 views

Compute the density of $Y=|X|$

When $X$ has the normal distribution $\mathcal N(\mu,\sigma^2)$ , compute the density of $Y=|X|$ I know ...
-1
votes
0answers
21 views

normal distribution variable plus not normal distribution variable

I just started to learn normal distributions and learned that a variable is normal if it has a pdf looks like : [1/(sqrt(2pi)*sigma)]*e^(-(x-mu)^2/(2*sigma^2). Now, i have X~N(0,1) and I need to ...
0
votes
0answers
5 views

Performance of an optimum estimator for Gaussian random variables used against Non-Gaussian random variables

Consider an optimum estimator for some parameter where the underlying distribution is following a Gaussian distribution with mean 'mu' and standard deviation 'sigma' (denoted by N(mu, sigma)). Let ...
0
votes
1answer
41 views

derive the mean and variance of $\bar X$ using means of sums rules

I can't find anywhere what the means of sums rules are so i'm confused with this question The random variables $X_1......X_5$ are jointly multivariate normal. Their expectations are $E(x)= \mu_i$ and ...
0
votes
0answers
47 views

Find the best predictor and the best linear predictor of $Y^2$ given $X$. Suppose $(X, Y ) \sim N(0, 0, 1, 1, p ).$

Once more, there's another question that I'm clueless on how to start. I should have dropped this course earlier. Suppose $(X, Y ) \sim BN(0, 0, 1, 1, p )$, meaning that $X$ and $Y$ are bivariate ...
1
vote
1answer
22 views

Calculate a probability involving drawings from bivariate normal variables with Xi and Yi i.i.d

There's a question which has been troubling me along with my earlier post. To be honest, I'm not entirely sure on how to proceed. All I know is that if X~N(mu,sigma^2) then P(X < A) = P(Z< ...
1
vote
1answer
24 views

Normal distribution probability function definition

Up to now, I believed that k-dimensional normal distribution has probability function: $\frac{1}{\sqrt{(2 \pi)^k |\Sigma|}}e^{-\frac{(x-\mu)^T\Sigma^{-1}(x-\mu)}{2}}$ Recently I have read an article ...
2
votes
0answers
31 views

Model selection: geometric mean of the standard deviation.

I have two models that represent a physical process. To determine which model is the best, I make some experiments and compare measured data with data predicted by each of the models. The model with ...
0
votes
1answer
14 views

Sum of variation for loads

The loads on an electrical network with 10 regions are modelled by considering a base load with mean 20mW and standard deviation 3mW. Variation due to regional load is modelled by considering that ...
0
votes
0answers
13 views

Sum of random variables for 2m tape

we use 2 metre tape for distance measurement and that the measurement error for the full tape length has 0 mean and variance 1.5cm^2. Find the mean and the variance if the total distance measured by ...
0
votes
0answers
24 views

Geometric Mean of Uniform random variables convergence

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
1
vote
0answers
26 views

The characteristic function of a multivariate normal distributed random variable

The characteristic function of a random variable $X$ is defined as $\hat{X}(\theta)=\mathbb{E}(e^{i\theta X})$. If $X$ is a normally distributed random variable with mean $\mu$ and standard deviation ...
0
votes
0answers
4 views

Expectation of log((w^Tx)^2) with respect to a multivariate Gaussian

I am interested in solving (or approximating) the following expectation. $$\int_x \mathcal{N}(x|\mu,\Sigma) \log((w^Tx)^2) dx$$ where $x,w\in R^D$ and $w$ is a constant vector. ...
1
vote
0answers
46 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
2
votes
2answers
38 views

Distribution of $U=\frac{X}{\| X \|}$ and $R^2 = \| X \|^2$ where $X=(X_1, \dots , X_n)$, $X_1, \dots, X_n \sim$ N(0,1) i.i.d. Independence?

I have the following problem: Let $X=(X_1, \dots , X_n)$, $X_1, \dots, X_n \sim N(0,1)$ i.i.d. What is the distribution of $U=\frac{X}{\| X \|}$ and $R^2 = \| X \|^2$. Are $U$ and $R^2$ independent? ...
2
votes
1answer
26 views

Add Chi-Squared Distribution to Normal Distribution

Let $z \sim N(\mu,\sigma)$. What is the distribution of $z^2+6z+1$?
0
votes
2answers
51 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 & ...
1
vote
1answer
23 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
42 views

What is the pdf of $Y =\min(X_1,X_2)$, considering lognormal random variables.

If $X_1$ and $X_2$ are two independent lognormal random variables with mean $m_1$ and variance $v_1$, mean $m_2$ and variance $v_2$, respectively. What is the pdf of $Y = \min(X_1,X_2)$?
0
votes
1answer
37 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 ...
1
vote
1answer
122 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
103 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 ...
1
vote
0answers
20 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
3answers
45 views

Computing standard deviation of discrete normal distribution

I used below pseudocode to generate a discrete normal distribution over 101 points. ...
0
votes
1answer
36 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 ...
1
vote
2answers
51 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
1answer
20 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] = ...
3
votes
2answers
79 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
1answer
37 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 ...
0
votes
2answers
32 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 ...
0
votes
0answers
176 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 ...
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
0answers
23 views

$(X, Y)^T$ ~ multivariate normal does NOT implies that X | Y $\in$ [a,b] has normal distribution??

I just found a rather surprising fact about the multivariate normal distribution. Suppose $(X,Y)^T$ has bivariate normal distribution; $$ \begin{bmatrix} X\\ Y \end{bmatrix} \sim MVN_2 \Big( ...
0
votes
0answers
34 views

Analytical computation of one Gaussian mixture model from another

I'm wondering if there is a way to analytically compute the optimal GMM (for a specific number of gaussians) in the case of approximating another GMM. E.g., is there an optimal single gaussian that ...
1
vote
1answer
35 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
83 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 ...
3
votes
1answer
113 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
52 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
0answers
22 views

Density of transformation of normal distribution

A data set contains real values $\left\{v_1,v_2,\text{...},v_k\right\}$, $k<\infty$. $X_n\sim \mathcal{N}(\mu ,\sigma ),\ n=1,2,...,k$ $P$ is the (not necessarily unique) permutation that ...