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

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-4
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0answers
13 views

so if I have a normal distribution with z=783 cm and sigma x = 150 cm and sigma y = 50 cm can I scale these sigmas for z=950? if so how? [on hold]

so I have a problem that says if I have a plane at z=783 cm, measure the sigma (standard deviation) of the distribution in the x and y directions. from the graphs projection in the y-axis, projection ...
0
votes
0answers
21 views

norma distribution and log-normal distribution

I often see when people analyzing data, they assume data has either normal or log-normal distribution, and trying to fit data into a distribution for the convenience of data analysis (e.g. by ...
1
vote
1answer
13 views

Chi-Squared Distribution

Let $Z_1, Z_2, Z_3$ be independent standard Normal R.V.'s. Which of the following has a Chi-Square distribution with 1 degree of freedom. $$ \begin{align} A) & & & \frac{Z_1^2, Z_2^2}{2} ...
1
vote
1answer
32 views

Solve for and Plot the Relationship Between Mean and Standard Deviation of a Normal Distribution Conditional on Satisfaction of A System of Equations

I am trying to use Mathematica, R, or Matlab to solve for (since it cannot seem to be solved analytically) and plot the relationship between mean and standard deviation of a normal distribution ...
-3
votes
0answers
14 views

Prove $\frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\rho^2}}$ [on hold]

Let $X_1,X_2$ have a bivariate normal distribution with zero means, unit variances, and correlation $\rho$. Show that $$ \frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\...
0
votes
2answers
38 views

Show that $(\bar{X})^2$ is not an unbiased estimator for $\mu^2$

If $X_1, ... , X_n$ are $n$ identical distributed independent random variables each with mean $\mu$ and variance $1$. A little confused by this question. Is it asking for if $(\bar{X})^2$ != $\mu^2$....
1
vote
0answers
18 views

Multivariate normal distribution conditional probability question.

$\newcommand{\Cov}{\operatorname{Cov}}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$We have that $X$ and $Y$ are random variables with a ...
0
votes
1answer
21 views

How many time the standard deviation, do I need to travel from mean in both directions such that I cover a given percentage of data?

I do not have much experience in Statistics. However, I read this rule on a page and followed it up on Wikipedia: https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule I wanted to know ...
2
votes
1answer
36 views

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to?

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to- $\frac{\Phi(1)-\frac{1}{2}}{\Phi(2)-\frac{1}{2}}$ $\frac{\Phi(1)+\frac{1}{2}}{\Phi(2)+\frac{1}{2}}$...
0
votes
0answers
10 views

How to obtain a unimodal histogram with normal distribution (gaussian)?

My task is to come up with a histogram consisting of $N$ bins. The histogram should show a (perfect) normal distribution. So something similar to what is shown in this image. How do I obtain the value ...
0
votes
1answer
16 views

Can we calculate the range form mean and standard deviation in a normal distribution?

Suppose in a normal distribution the mean is 90 and the standard deviation is 10. Then what is the range? Would the following be an acceptable way to find the range, where $\sigma$ represents the ...
1
vote
1answer
31 views

Integral of a line with random gradient

Consider a random line $Y = Mx$ where $M$ is a standard normal variable $M \sim \mathcal{N}(0,1)$. The line is integrated between 0 and 1: $$I = \int_{0}^{1} Y dx = \int_{0}^{1} Mx dx$$ What is the ...
0
votes
0answers
16 views

Non integer, non-centered Gaussian moments

I have read the following question : Non-centered Gaussian moments where it is stated that : $$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{...
0
votes
1answer
45 views

normal distribution formula conventions

I sometimes see people write normal distribution formula like this, wondering if $G$ means Gaussian? And what does $C$ means here? Thanks. $G(\mu, \sigma)$ $\exp(\mu + C(\sigma))$ thanks in advance,...
0
votes
3answers
81 views

How can I compute $\mathbb{E}[Z^4]$ where $Z\sim N(0,1)$

Let $Z\sim N(0,1)$ and $Y=a+bZ+cZ^2$. I want to compute the variance of $Y$. This is what I did: $$\operatorname{Var}(Y)=0+b^2\operatorname{Var}(Z)+c^2\operatorname{Var}(Z^2)=b^2+c^2\operatorname{Var}...
2
votes
2answers
30 views

Convolution of normal distribution not equal to product with constant?

Convolution of a normal distribution says: If, $X \sim \mathcal{N}(\mu, \sigma^2)$, then $X+X\sim\mathcal{N}(\mu+\mu, \sigma^2+\sigma^2)=\mathcal{N}(2\mu,2\sigma^2)$ However, Multiplication of a ...
0
votes
1answer
33 views

Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$

Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$ I tried solving this in 2 different ways. The second way is what I am stuck on: 1st Way: Let $Y=e^X$ where $X \sim N (\mu,\sigma^2)$. ...
0
votes
1answer
29 views

Proof that normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = (\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \frac{...
0
votes
1answer
50 views

confused about proposal distribution in MCMC

This is a question from notes I have some questions regarding the proposal distribution which is $N(x,1)$ Is the proposal distribution symmetric i.e. $g(x_p|x)=g(x|x_p)$? I'm not sure whether it ...
4
votes
1answer
1k views

Inverse Mills ratio for non normal distributions.

We have the well known result of the inverse Mills ratio: $$ \mathbb{E}[\,X\,|_{\ X > k} \,] = \mu + \sigma \frac {\phi\big(\tfrac{k-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{k-\mu}{\sigma}\...
2
votes
1answer
48 views

Question about multidimensional iid random variable

Let $X=(x_1,\ldots,x_d)^\top\in[0,1]^d$ be the row-wise representation of an $n\times n$ image ($d=n\times n$). Each element of $X$ is the value of a pixel, which we assume it belongs to $[0,1]$. If ...
0
votes
0answers
20 views

Characteristic function of standard normal distribution using this method.

Lets have $f(t)$ be this characteristic function. I am told that $f'(t)=-t \cdot f(t)$ and that this can be proven, I found using partial integration and the dominated convergence theorem. I am aware ...
0
votes
1answer
62 views

Uniform Distribution / Normal Distribution

Let the random variable X ~ U ( 0, k ) and Y is a second random variable such as Y | X ~ N ( X , 1). a) Determine the Y density function if k = A . b) Determine the value of k if COV [X , Y ] = B. a) ...
0
votes
1answer
21 views

Assumption of normality while creating CI from chi-squared and t-statistic pivots?

While explaining the use of a chi-squared pivot or a t-statistic in creating confidence interval, we were told that one of the underlying assumption is the normality of the data. Chi-squared ...
0
votes
0answers
7 views

Ga algorithm for MLE of normal denstity [closed]

I want to use GA package instead of optim or other packages in R for finding mle of normal distribution numerically. I want to bsimmulate from this distribution,find and minimize MSE of estimations. ...
0
votes
0answers
25 views

Help with normal distribution

Mary only has one lamp in her house, and the lamp only has ONE light bulb. She has bought 50 light bulbs, where each and everyone of them has an exponential distributed lifetime, of μ = σ = 1500 hours....
0
votes
1answer
24 views

Simplifying $\int_{-\infty}^z \phi(x)\,\Phi(\beta\, x)\,dx$, $\phi(x)$ pdf of normal, $\Phi(x)$ CDF of normal

Can we simplify further the following function? $\int_{-\infty}^z \phi(x)\,\Phi(\beta\, x)\,dx$, Where $\phi(x)$ is the pdf of standard normal distribution, i.e., $\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-\...
0
votes
1answer
40 views

Expectation of absolute random variables with mean 1 and standard deviation 1

For a random variable $\gamma \sim \mathcal{N}(\mu,\sigma)$ , were is $ \mathcal{N}$ is the normal distribution. What is the way to calculate the following: $ \mathbb{E}[|\gamma|] = ? $ And ...
1
vote
0answers
55 views

Difference of Entropy of two-dimensional Gaussians

I encountered a putative contradiction. Assume we have two 2-dim. Gaussian variables $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$ with all components being independent, normal distributed variables: $x_1,...
2
votes
1answer
50 views

What is the chance a team will have at least 10 more wins than losses at any point in a 100 game season? They have a 50% chance of winning each game.

More generally: Each game, $n = 1,2,...,N$, a team has probability, $p = 0.5$, of winning. Their standing $x$ is given by $x(n) = x(n-1)\pm1$ depending on whether they win ($+1$) or lose ($-1$). Their ...
1
vote
0answers
20 views

I need help normalizing a Gaussian kernel matrix to integer values

I am trying to understand the mathematics behind Canny edge detection, and the first step is to apply a Gaussian blur to the image you are working with. To do a Gaussian blur, you must obtain a ...
2
votes
0answers
41 views

normal distribution hazard rate increasing function

How to show this function is increasing convex function: Define $f(z)=\frac{T(z)}{g(z)}$, where $T(z)=\phi(z)-\alpha \phi(\frac{z}{\alpha})+z(\Phi(z)-\Phi(\frac{z}{\alpha}))\,,$ $g(z)=\Phi(z)-\...
0
votes
0answers
9 views

Relationship between complex normal and bivariate normal distributions

Suppose I have a complex random variable $X$ which follows a complex normal distribution (with $0$ mean). I've been trying to represent the complex normal in a simpler way, but I'm not sure how. Is ...
1
vote
1answer
25 views

How to find normal distribution that has a quadratic?

Let $X$ be a normal random variable with mean 1 and variance 4. Find $P(X^2 − 2X ≤ 8)$. (Answer key .86) My attempt $$P(X^2-2X\le 8)=P((X+2)(X-4)\le 0)$$ and this is where I am lost. I did the ...
2
votes
0answers
20 views

Means and Covariances of powers of a normal distribution

Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T $$ What is the expected ...
1
vote
1answer
35 views

Having trouble understanding the output of the integral

Exposure is given by $$E=\max(V,0)=\max(\mu+\sigma Z,0)$$ The EE defines the expected value over the positive future values and is therefore:$$\mathbb{E}[E]=\int_{-\mu/\sigma}^{\infty}(\mu+\sigma x)\...
0
votes
0answers
11 views

what are the mean vector and covariance matrix of the multivariate truncated normal distribution?

Let $\mathbf{X}=(X_1,X_2, \ldots, X_p)^\prime$ has multivariate normal distribution with mean vector $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$, i.e. $\mathbf{X}\sim N_p(\mathbf{\mu}, \...
0
votes
1answer
27 views

Finding probability sample proportion is less than 33% assuming null hypothesis is true

Candidates 1,2 and 3 are running for a position in a company. Candidate 1 claims 38% favourability among all the voters. Assuming this is true, what is the probability that in a random sample of 500 ...
3
votes
1answer
31 views

Asymptotic Moments of the Binomial Distribution, $E(X/(np))^k = 1 + O(k^2/n)$?

Let $X \sim \text{Binomial}(n, p)$ be the sum of $n$ Bernoulli($p$) random variables. What is the value of $E(X/(np))^k$, where $k$ is a large integer, as $n$ grows large? From calculations the ...
1
vote
0answers
21 views

Build skew normal distribution knowing the mean, max, and min

Say I have a data point with included errors and I want to build some continuous distribution around it. Normally this might be a Gaussian because one knows the sigma and mean right off the bat. ...
0
votes
1answer
26 views

Testing the independence of two jointly normal variables

Variables $u$ and $v$ are jointly normal, correlated with zero mean. $X$ is a linear combination of $u$ and $v$: \begin{align*} X := \frac{u}{\sqrt{E(u^2)}}-\rho\frac{v}{\sqrt{E(v^2)}} \end{align*} ...
1
vote
1answer
46 views

Nomrmal probability distribution

A nationalised bank has found that has dialy balance available in its savings accounts follows a normal distribution with a mean of rs. 500 and standard deviation of rs. 50 The percentage of saving ...
1
vote
1answer
28 views

Why a normal distribution would not give a good approximation to the distribution of marks

An examination is marked out of $100$. It is taken by a large number of candidates. The mean mark, for all candidates, is $72.1$ and the standard deviation is $15.2$. Give a reason why a normal ...
0
votes
2answers
20 views

Mean Absolute Deviation of normal distribution

The Mean Absolute Deviation of the normal distribution is simply $$\sqrt{\frac{2}{\pi}}\sigma,$$ where $\sigma$ is the standard deviation of the normal distribution. (Wikipedia, Mathworld.) How do I ...
0
votes
0answers
16 views

How do you create a Gaussian distribution on a polynomial ring?

In the specification for the YASHE homomorphic encryption algorithm, it says that for the parameter generation subroutine, you need to: Given the security parameter $λ$, fix ... distributions $\...
1
vote
0answers
18 views

Marginal conditional mean of two dimensional Brownian motion, using more than one time point.

EDIT: I found the error! I do not think this question is relevant for anyone, but I cannot find out how to delete it - please feel free to if you have the read this and have the option. I have ...
0
votes
2answers
65 views

What is the expected distance between normally distributed points on a plane? What about the distance in higher dimensions?

Let $X = (x_1, x_2)$ and $Y = (y_1, y_2)$ where the random variables $x_1$, $x_2$, $y_1$, $y_2$ are independent standard normal. What is the expected distance between $X$ and $Y$, i.e. what is $$D_2=E\...
1
vote
0answers
23 views

Double Integral of normal cdf and gamma pdf

I am trying to solve the following double integral for a problem that came up during my research: $$\int_0^{\infty}\int_0^q \frac{1}{q} \Phi\left(\frac{u-\mu l-kQ}{\sigma l}\right)\mathrm du f_L(l)\...
0
votes
0answers
25 views

How to rotate an $n$-dimensional normal distribution, to maximize the likelihood of a sample

Suppose we have a normal distribution with a diagonal covariance matrix S and mean $0$, i.e. $N(0,S)$. I want to find a Rotation matrix $R$, to rotate this distribution to maximize the likelihood of a ...