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

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2
votes
1answer
257 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
1
vote
0answers
5 views

Normal distribution and option pricing [on hold]

I have a question: A random variable $z$ is normally distributed. Two options have a strike price $k$, then which has a higher value $|z|$ or $z^2$? Thanks for the help!
0
votes
1answer
14 views

Why normal approximation to binomial distribution uses np> 5 as a condition

I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success. On most websites it ...
0
votes
0answers
12 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
-2
votes
0answers
17 views

Normal distribution exercise [on hold]

In a factory, compacts are filled with a cosmetic powder. We consider the weight of the powder follows a normal distribution $N\sim(\mu, 1.21)$. The value of $\mu$ depends on the setting of the ...
1
vote
2answers
48 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
3
votes
0answers
21 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 $F_{Y_n}$ ...
4
votes
1answer
234 views

Confusion related to integral of a Gaussian

I am a bit confused about calculating the integral of a Gaussian $$\int_{-\infty}^{\infty}e^{-x^{2}+bx+c}\:dx=\sqrt{\pi}e^{\frac{b^{2}}{4}+c}$$ Given above is the integral of a Gaussian. The ...
0
votes
0answers
84 views

PDF of X +Y + X* Y, when X and Y are independent Normal [on hold]

I have $X,Y$ iid Normals $N(0,\sigma^2)$ What is the distribution of $X+Y+YX$? Thnks a lot!
0
votes
1answer
62 views

Statistics: Relationship between process capability and mean

A company produces one-kilogram sugar packets. The specifications on the net content are 1000 ≠ 5 grams. Assuming that the net content follows normal distribution with mean weight as 1005 grams and ...
4
votes
2answers
93 views

Fraction Problem. 3rd grader question got parents thinking

So our nine year old son comes home from 3rd grade and tells us an amazing thing happened in school today. He was playing a math game with his friend and they got the same score two times in a row! ...
0
votes
2answers
41 views

Normal distribution squared probability

Let $X_1,X_2,X_3,X_4$ be independent standard normal random variables and $Y=X^2_1+X^2_2+X^2_3+X^2_4$. Find the probability that $Y≤3$. Enter your answer as a decimal and make sure that at least $10$ ...
1
vote
1answer
41 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
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
9 views

how to find the value of n of a normal distribution in R

I have an example that my variable Y is the number of hours (eating say) which follows a N(50,8) distribution. Is there any way to find the number of hours that follows this distribution? Thanks
0
votes
0answers
19 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) ...
-2
votes
0answers
11 views

Confidence Interval Estimate [on hold]

Assume a simple random sample is taken, the conditions for a binomial distribution are satisfied, and the sample proportions can be approximated by a normal distribution. From a sample of $200$ fish ...
1
vote
1answer
25 views

Mean and variance: Gaussian is the most conservative assumption

"given only the mean and variance of a distribution, the most conservative assumption that can be made about the distribution is that it is a Gaussian having the given mean and variance" I've read ...
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 ...
0
votes
1answer
18 views

stdev and mean from gaussian fit vs. from classical formula

I have a set of data - measured speed of molecules in water. I made a histogram and fitted it with function $$A\exp\frac{(x-B)^2}{C}$$ calculating mean and standard deviation from values B and C If I ...
1
vote
0answers
19 views

Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
0
votes
0answers
11 views

Pivotal quantity that is a function of the z-score: find the CI

** Assumptions ** Let: $X$ be a random variable. $\bar{X}_n$ be the sample mean of X; $\mu$ be the expectation value of X; Assume that $\mu$ is not observable; $S_n^2$ be the sample variance of ...
2
votes
3answers
3k views

Standard deviation of the weighted mean

How do you find the standard deviation of the weighted mean? The weighted mean is defined: $\bar{x}_w = \frac{\sum{wx}}{\sum{w}}$ The weighted standard deviation (since it is not specified, I take ...
1
vote
2answers
205 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
0
votes
0answers
15 views

finding probabilty of x=? on z table

I was given a question: what is the probability that $x$ is less than $3 P(X<3)$ This is normal distribution mean $= 2.4 SD = 1.3856$ In the working they said to do the following: $P(X<3) = ...
1
vote
1answer
420 views

probability of sample variance lying between given values

Let $X_1,\ldots,X_n$ be a random sample of size $n = 10$ from a population which is Normally distributed with mean $= 48$ and variance $= 36$. What is the probability that the sample variance of such ...
1
vote
1answer
360 views

Lognormal definition. Clarity please?

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
0
votes
0answers
17 views

When $(X, Y)$ are jointly normal given that both $X$ and $Y$ are normal?

We know that $X$, $Y$ are normal does not guarantee $(X, Y)$ is jointly normal. A typical example is: $X=Z$, and $Y=ZU$, where $Z$ standard normal, $P(U=1)=P(U=-1)=1/2$, and $Z, U$ are independent. ...
1
vote
0answers
18 views

Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
3
votes
1answer
262 views

How to plot standard deviation rings of a bivariate normal distribution?

I'm working on a project right now where I have Gaussian distributions, and I want to create a graphic that represents them. I'm not sure how to generate the ellipse that represents say 1 standard ...
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 ...
8
votes
1answer
219 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two n-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
0
votes
1answer
11 views

Geometric Sequence with Normal Distribution Problem

Given: The running time (in seconds) of an algorithm on a data set is approximately normally distributed with mean 3 and variance 0.25. a. What is the probability that the running time of a run ...
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
2answers
26 views

normal distribution using Z - finding probability between 2 numbers

I am wanting to find the probability of the following: SD = 20 Mean = 100 P(85 < X < 117) i have found the z values for both: P(X>85) : X-u/o = 85-100/20 Z = -0.75 and found the ...
0
votes
2answers
19 views

Understanding sampling from a normal distribution with zero mean

I'm studying probability. I came a cross "sampling from distributions". Given a probability density function $f_X(x)$, what I understood is that sampling means getting values of $x$ according to the ...
0
votes
2answers
28 views

Normal Distribution Problem

The time taken for a computer to connect to a server is normally distributed with a mean value given by 3.3 seconds and a standard deviation of 0.66 seconds. (a) A computer is said to have a fast ...
1
vote
1answer
203 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
1
vote
1answer
37 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
0
votes
3answers
71 views

Gaussian integral evaluation

Asked a question to evaluate the Gaussian Integral, $$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx $$ using the the following approximation, $J=\Bbb E[X^2] \sim J_N = 1/N ...
1
vote
0answers
29 views

Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous?

Let $X$ be a standard Gaussian random variable. Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous ? I don't understand the question here. Now $X$ has density ...
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: ...
0
votes
1answer
40 views

Expectation formula proof [closed]

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

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
-1
votes
0answers
9 views

how to use standard normal table?

I don't get it. How do I use the following "type" of table? I can use the ones with negatives just fine, but I don't understand this one: ...
2
votes
2answers
51 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 ...