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

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0
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1answer
29 views

How to determine the multivariate distribution?

Consider $$ Z_1:=\bar{Y}_1-\bar{Y}_2\sim N(0,\sigma^2(n_1^{-1}+n_2^{-1})),\\ Z_2:=\bar{Y}_1-\bar{Y}_3\sim N(0,\sigma^2(n_1^{-1}+n_3^{-1})),\\ Z_3:=\bar{Y}_2-\bar{Y}_3\sim ...
1
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1answer
34 views

Show that $E(S)=\sqrt{\frac{1}{n-1}}\frac{\Gamma(n/2)}{\Gamma[(n-1)/2]}\sigma$

Let $X_1,...,X_n$ be a random sample of size $n$ from the normal distribution with mean $\mu$ and variance $\sigma^2$ and let $S^2=\frac{1}{n-1}\sum^n_{i=1}(X_i-\bar{X})^2$ be the sample variance. ...
1
vote
1answer
55 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
2
votes
2answers
33 views

Distribution of a sum of normal distributions?

$X$ = weight of a small bag of crisps has Normal distribution with mean = $35.5$ and $var = 0.8$ . $Y$ = weight of a large bag of crisps has Normal distribution with mean = $152$ and $var = 3.2$ ...
0
votes
1answer
21 views

Normal Random Variables

Let Z1 and Z2 be independent standard normal random variables. What is the probability that the minimum of Z1 and Z2 will be greater than 1.0? How do I go about this when I have no values? Is the ...
-2
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1answer
23 views

Normal Random Distributions

A university expects to receive, for the next academic year, 16000 applications for admission to the bachelor’s degree program. The SAT score obtained by the applicants is modeled as a normal random ...
2
votes
1answer
52 views

Combining statistical distributions

I have a situation where a distribution is dependent on 2 variables, one of which follows the poisson distribution, and the other the normal distribution, and I want to establish the method of ...
-4
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1answer
38 views

If a random variable X has mean of μ and standard deviation σ…

, then what will be the mean and standard deviation of (X − μ)/σ ?
1
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0answers
27 views

Can this be solved analytically?

I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is $f(x) = 2 C_1 (X_1 - x) ...
4
votes
1answer
137 views

Why does this determinant have a continuous density at zero?

This question is a simplification of my previous question. I think this is easy, but I don't have a strong enough background in probability. Let $A$ be a random $n\times n$ real matrix that satisfies ...
2
votes
1answer
29 views

$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$ independent and normal

I am looking for a nice proof of the following statement: If $X,Y\sim U(0,1)$ are two independent uniformly distributed random variables, then $$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), \quad ...
0
votes
0answers
23 views

GEV of Normal Distribution and relationship of the parameters

My question goes on Extreme Value Theory for the Normal distribution: 1) Which GEV (Generalized Extreme Value distribution) type is the Normal distribution(Weibull/Gumbel/Frechet)? 2) If we have the ...
0
votes
1answer
30 views

How to add standard deviation regarding MATLAB function normrnd(mu,sigma)?

My question depend on this scenario which is as follows, I used a MATLAB function "normrnd(mu,sigma)" with mean 'mu = 0' and S.D 'sigma = 5', to generate a normal random number "R1". I added this ...
0
votes
1answer
44 views

Geometric BM tends to zero but is strictly positive a.s.?

The process $\{S_t\}_{t\ge0}$ following $dS_t = \sigma S_tdW_t$ with $S_0>0$ has the solution $$S_t=S_0 e^{-\frac12\sigma^2t+\sigma W_t}$$ Now for any $\epsilon>0$ we have $$\mathbb ...
2
votes
1answer
52 views

if $X_i$ are iid standard normal distributed, what is the limiting distribution of $\sum X^4 / (\sum X^2)^2$?

If $X_i$, $i=1,\ldots,n$ are iid standard normal distributed, what is the limiting distribution of $S_n=\sum X^4 / (\sum X^2)^2$? After finding the moments and since $Cov(X^4, X^2)=0$, I have the ...
0
votes
0answers
36 views

Problem on Expectation

Let $\Phi$ denote the standard normal distribution. Suppose $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. Show that ...
0
votes
1answer
36 views

Approximation in Normal distribution random variable

Let ${X_n : n \geq 1}$ be independent $\mathcal{N}(0,1)$ random variables. How do we get the following approximation?
0
votes
2answers
33 views

Given N-distribution, calculate expected value and Var of a function

$X_{{i}}=X_{{1}}...X_{{n}}$ is an iid. random variable with the distribution $N \left( {\frac {\alpha}{\beta}},{\frac {{\alpha}^{4}}{{\beta}^{4}}} \right) $ I would like to calculate the expectation ...
2
votes
0answers
22 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
0
votes
1answer
40 views

Find the probability that the average of X and Z is greater than Y. Where X, Z, and Y are normal RVs.

Here is the exact statement: Suppose X,Y , and Z are independent random variables. X is a normal random variable with mean 5 and variance 16, Y is a normal random variable with mean 7 and variance ...
0
votes
0answers
7 views

Moment generting function-Correlated Normal Variables

I have a variable x which is normally distributed with mean u1 and standard deviation sigma 1. We have another variable Y which is normaly distributed with mean u2 and standard deviation sigma 2. The ...
1
vote
1answer
39 views

Cumulative distribution function of a degenerate multivariate normal distribution

Let $X\in\mathbb{R}^{n}$ be a multivariate normal variable with the mean vector $\mu$ and the covariance matrix $\Sigma$. It is well known that if the matrix $\Sigma$ is positive-definite the ...
1
vote
2answers
27 views

MLE of MVN($\mu, \Sigma$)

I'm trying to find MLE of MVN($\mu, \Sigma$), i.e $N_k(\mu, \Sigma)$ with random sample $X_i, 1\le i \le n$. It was easy to get $\widehat{\mu}= \bar{X}$ and $\hat{\Sigma} = \frac{1}{n} \sum_i (X_i - ...
0
votes
2answers
31 views

Sum of maximum of two correlated normal random sequences

Let $x_{1},x_{2},\cdots,x_{n}$ and $y_{1},y_{2},\cdots,y_{n}$ be correlated normal random variables the covariance between two arbitrary random variables is known. In other words, let ...
1
vote
2answers
49 views

Concept of Probability in math first level

I am trying to teach myself the concepts of probability and I was wondering if this is correct. I am only 13 years old and did not learn this yet. I am just reading parts of a probability book to get ...
1
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1answer
15 views

How to calculate Standard deviation with mean 0 and Min and max value on x-axis is -1 and 1 respectively?

How to calculate Standard deviation with mean 0 and Min and max value on x-axis is -1 and 1 respectively? It is of-course a normalize distribution. I apologize in advance for stupid question.
3
votes
1answer
98 views

Linear combination of normally distributed variables

We know that if $X \sim N_p(\mu, \Sigma)$ then $a'X \sim N(a'\mu,a'\Sigma a)$ for and $a \in \mathbb{R}_p$. What I need to know is if the converse of this is also true. Can this be proved? Would ...
0
votes
1answer
44 views

Why is this multivariate $3\sigma$ ellipse rotated?

While reading this answer, I clicked on the provided link to this Wikipedia page. The main article image shows the PDF of a 2D multivariate normally distributed system: In the image, the $3\sigma$ ...
-2
votes
1answer
64 views

I would like some help please in utilising the normal distribution.

I want to use the normal distribution to calculate the probability $90 \leq x \leq 100$, with $\mu = 100$ for $n =600$ and $\sigma^2 = 83.333$. Now I think this means $\frac{90 - 100}{\sqrt{83.333}} ...
0
votes
2answers
39 views

“Show experimentally” that for large $N$, $X$ appears to be normally distributed.

I'm a bit confused about the following problem: Let $X$ be the random variable $$X = \frac{X_1+X_2+...+X_N}{\sqrt{N}}$$ where $X_k$ is the outcome from the $kth$ flip of a fair coin where heads ...
1
vote
2answers
83 views

Sum of two truncated gaussian

What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a ...
1
vote
2answers
43 views

An exercise about Borel paradox

If $X$ and $Y$ are independent standard normals, what is the conditional distribution of $Y$ given that $Z=1$, where $Z=I(X=Y)$?
1
vote
1answer
18 views

“fractional” expectation of zero-mean normal distribution

I'm trying to calculate $E[X^{\frac{2}{3} } ] $ of a zero-mean normal distribution. Any help to solve $$ E[X^{\frac{2}{3} } ] = \int\limits_{-\infty}^{\infty} x^{\frac{2}{3} } \frac{1}{\sigma ...
1
vote
1answer
24 views

How to get a Gaussian curve fitting a given range of values?

I was trying to find a way to make a gaussian function out of a range of values: $1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\ 15\ 16$ I want the mean to be the most probable value, $8$ and the ...
1
vote
1answer
16 views

Integrating the error function in a calculation related to Brownian motion

I wish to calculate the probability that a standard linear Brownian motion $B(t)$, $t\ge 0$, will be at time $t_0$ inside the interval $[a,b]$, and at time $t_1$ in the interval $[c,\infty)$. To do ...
1
vote
0answers
55 views

Conditional density based on 2 gaussian measurements

However intuitive, I don't understand the formulas for the conditional mean and variance from 2 gaussian measurements. I have not found anything relevant mainly because I don't think I'm searching ...
1
vote
2answers
41 views

Normal distribution, how to calculate $\mu$ and $\sigma$

How to calculate $\mu$ and $\sigma^2$ when it is known just that $P(X\le 49)=0.6915$ and $P(X>51)=0.2266$ ? Thank you very much!
2
votes
1answer
23 views

Normal Distribution burnout… of lightbulbs.

Thank you for looking through this problem, much appreciated! I tried to work out the answer for a, but I got .2946 when the actual answer is .3085... How do I start this? By the way, I just want to ...
2
votes
1answer
20 views

How do I show the covariance matrix of a multivariate normal random vector is positive definite?

The question is as follows: Suppose the $n$-dimensional random vector $\textbf{Z}$ has mean vector $\mu$ and variance-covariance $V$. By considering $Var(x^{T}\textbf{Z})$ for $x \in \mathbb{R}^n$, ...
0
votes
1answer
31 views

Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $E[||x||_2]$, $x $~$ ...
0
votes
0answers
12 views

Two i.i.d Rvs (Gaussian)

Q: You have two i.i.d Rv's X~N(0,1) Y~(0,1). Let Z=(X+Y)^2. a) Find the mean on Z i.e E[Z}. b) Find Corr(X,Z) & Corr(Y,Z). c) Determine if Z & Y are uncorrelated. Ans: Finding E[Z] was ...
1
vote
2answers
36 views

The Normal Distribution in measuring two towers…

I understand the explanation and the math behind the problem, all I am asking for is a quick explanation behind this. "Two instruments are used to measure the height, h, of a tower. The error made by ...
0
votes
1answer
7 views

Bivariate Normal Probability

Assume we have a large data set of PSAT and SAT scores with bivariate normal distribution with $\rho = 0.6$. The mean and SD of the PSAT scores are (respectively) $1200$ and $100$. The mean and SD ...
0
votes
1answer
16 views

$\mathbb{P}(|X|<1,|Y|<2)$ When $X,Y$ Are I.I.D. Standard Normal

Calculate $\mathbb{P}(|X|<1,|Y|<2)$ when $X,Y$ are i.i.d. standard normal r.v.s. I think the answer is simply $$(\Phi(1)-\Phi(-1))(\Phi(2)-\Phi(-2)).$$ Is this correct? Thanks.
0
votes
1answer
22 views

An IB Math HL question on normally distributed random variable.

Some Background: Tim goes to a popular restaurant that does not take any reservations for tables. It has been determined that the waiting times for a table are normally distributed with a mean of ...
0
votes
1answer
10 views

Distribute range of score among objects

I need some help with the following. I have 10 or X amount of subjects with a rating and would like to distribute a score of 1 to 5 between them based on their rating. The subject with the highest ...
1
vote
1answer
41 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
0
votes
1answer
21 views

Model going from Normal to Log-Normal

I'm getting in a real mess at the moment over something I think is very simple, as well as the wording/terminology. I have a model - $\ln(Y(x))=a+b\ln(x)+\epsilon, ...
0
votes
1answer
25 views

Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= ...
0
votes
0answers
25 views

Maximisation of Conditional Gaussian Mixture Model using EM Algorithm

Assume, the pdf of conditional Gaussian mixture distribution of $X_{A}$ given $X_{B}$ is formulated as follows: $f(X_{A}/X_{B}) =\sum^{K}_{k=1} ...