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

learn more… | top users | synonyms

1
vote
2answers
36 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
36 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
56 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
vote
1answer
35 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
103 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
54 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
253 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
40 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
99 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
46 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
21 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
27 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
21 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
42 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
27 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
23 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
41 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
16 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
51 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
8 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
34 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
49 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
24 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
35 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
33 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} ...
0
votes
0answers
24 views

For which joint distributions is a conditional expectation an additive function?

I know that, for a random vector $(X,Y,Z)$ jointly normally distributed, the conditional expectation $\mathbb{E}[\,X\mid Y=y,Z=z]$ is an additive function of $y$ and $z$. For what other distributions ...
0
votes
2answers
80 views

Distribution of mean of Normal distribution

Suppose $X\sim N(\mu,\sigma)$. I want to find the following probability $P[\mu \ge \theta |x= \theta -c]$ for $c>0$. In another word, I saw a sample of Normal distribution, $x$, and know that it ...
0
votes
1answer
20 views

Does correlation have to be in the context of (Gaussian) normal distribution?

I am not quite familiar with the concept of correlation. The Pearson's correlation coefficient is defined as: $\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ...
0
votes
0answers
50 views

Expectation of maximum of product of normal random variables

Let $X_i Y_i \sim N(0,\sigma^2) N(\mu,\sigma^2b)$, $\mu \neq 0 ,b >0$. then is there any inequality for the maximum of these products. What I mean is $E(\max{X_iY_i, 1 \leq i \leq m})$. I have ...
0
votes
1answer
17 views

Show that $Z-\tilde{Z}_{\iota_{\nu}}$ and $\tilde{Z}$ are independent.

Let $Z\sim N(a\iota_{\nu},I_{\nu}), a\in\mathbb{R}$ whereat $$ \iota_{\nu}=\begin{pmatrix}1\\1\\\vdots\\1\end{pmatrix},~~~I_{\nu}=\text{diag}(1,\ldots,1). $$ Show that ...
1
vote
0answers
53 views

Bivariate normal distribution; rotation; diagonal covariance matrix

Let $Z\sim N(0,\Sigma)$ with $$ \Sigma=\begin{pmatrix}\sigma_1^2 & p\sigma_1\sigma_2\\p\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix} $$ whereat $\sigma_i^2=\text{var}(Z_i), ...
0
votes
0answers
14 views

Controlling a random variable

I've got a system the output of which is a random variable with a certain distribution, which for the purpose of this discussion can be assumed to be normal. The input variable is voltage. The ...
1
vote
0answers
17 views

Binomial distribution vs Normal distribution

It is often said that the normal distribution "approximates" the binomial distribution. What is the precise mathematical expression of this fact?
3
votes
0answers
44 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that ...
0
votes
0answers
43 views

How to compute the expected value of normal distribution over a finite interval.

The occurrence time of event A is normally distributed with mean $\mu=200$ and variance $\sigma^2=10^2$. That is, $f(A) \sim \mathcal{N}(200, 10^2)$. As known, the expected occurrence time of A can ...
1
vote
0answers
16 views

Expectation of product of Normal CDFs w.r.t. a bivariate Normal distribution?

I am trying to figure out if there is a closed form expression for the following expectation: $\int\int \phi(\gamma_1)\phi(\gamma_2) \mathcal{N}(\gamma\big|\mu, \Sigma)d\gamma_1 d\gamma_2$ where ...
0
votes
1answer
73 views

Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
0
votes
0answers
54 views

Expectation of normal CDF with truncation

Suppose that $a$ and $T$ are given positive numbers. I would like to evaluate $$\begin{align*} \mathbb{E}\left[\Phi\left(aX\right)\mu\left(X+T\right) \right],\tag{1} \end{align*}$$ where ...
1
vote
1answer
33 views

Normal distribution without standard deviation given

The proportion of pink candies in a bag is supposed to be $50\%$. The filling machine is to be tested to see if it fills with the right proportion. A random sample of $50$ candies is made. The machine ...
1
vote
0answers
14 views

Modeling Gaussian Error

Context I am designing a simulation of a robot receiving input from a sensor which has gaussian error. The robot will start from a known position and move at a constant speed; the sensor will ...
1
vote
2answers
39 views

When do normal distributions not occur?

I know that in many cases one can assume a normal distributed probability density. But what the situations when the distribution in non-normal. Some examples would be nice. For example, suppose ...
0
votes
0answers
24 views

Estimation for expected value of a combination of two normal distributed random variables

I am struggeling to understand the proof of the following Lemma. Let $\epsilon_1$, $\epsilon_2$ be $\mathcal{N}(0,1)$ random variables. Then $\forall$ b $\geq$ 0 and $\forall$ c $\geq$ 1 there is an ...
4
votes
1answer
140 views

Probability of a gaussian distribution in another gaussian distribution

Assume we have a Gaussian distribution $p(x) \sim \mathcal{N}(\mu_p,\Sigma_p)$ For any point $X$, it is easy to compute the density of $x$ in $p$: $$p(x) = \frac{1}{|2\pi ...
3
votes
0answers
62 views

Independent normal distributions

I found two theorems with a similar content and want to find out which one is true: Let $X,Y$ be normally distributed random variables and $X+Y$ is also normally distributed or $ (X,Y)$ is ...
1
vote
0answers
57 views

How to use the normal probability table in reverse

I'm just wondering if anyone could give me a bit of advice on this. This relates to CCEA's S1 exam questions. $Z \sim \text{N}(0, 1)$ Let's say $\phi(z) = 0.5015$ Find z. Here is an extract of the ...
1
vote
1answer
43 views

moment generating function of normal distribution

I know this question relates to the chi-squared distribution, but I think what the question wants me to do is somehow derive this distribution from the information given. I have a normally ...
2
votes
1answer
66 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...