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

learn more… | top users | synonyms

1
vote
1answer
6 views

Standard Normal Distribution Findng A

I have the following question and i am dumbfounded on how to find the a in my given question. $$\sigma= 10000$$ $$\mu= 50000 $$ Find the monthly income which is exceeded by 10 % of employees. I ...
0
votes
1answer
15 views

Independent and identical

I have $X_1,X_2, \cdots X_n$ that are iid from $N(\mu,\sigma)$. In my derivation of the expression $E \big( \sum^n_{i=1} X_i^2 \big)$ I have written $E \big( \sum^n_{i=1} X_i^2 \big) = \sum^n_{i=1}...
0
votes
2answers
34 views

How to find range when mean and standard deviation is given in a normal distribution of 100

The earnings of 100 workers in a company are normally distributed. if the mean is 24 and standard deviation is 4, find an approx value for the range Edit 1. Your help is much appreciated guys. But ...
0
votes
0answers
12 views

Creating Standard Normal Distribution Curve with 26 years of Price Data

I am attempting to create a standard normal distribution and graph it with 26 years of price data from Crude Oil. I've managed to create the graph; however, i'm not sure if it's correct and it's not ...
1
vote
3answers
30 views

Jointly normal and correlated normal random variables

Is is true that if two normal random variables are correlated, then they are jointly normally distributed? I am not sure how to prove or disprove it.
0
votes
1answer
56 views

Poisson distribution with more than one lambda.

An archaeologist has two old pieces of wood and shall decide which piece of wood is the oldest. Radioactivity from the pieces of wood are recorded by a counter. Number of registrations per unit time ...
0
votes
1answer
25 views

Finding the Intersection of Normal CDF and y=x and Plot the Relationship Between $\sigma$ and Intersection Conditional on Different $\mu$

I'm trying to investigate the intersections of the Normal CDF and the $y=x$ line for $\mu \in [0, 1]$ and $ \sigma \in (0, 1] $ when $ x \in [0, 1] $ and plot the corresponding relationship between $\...
0
votes
0answers
12 views

resource for derivation showing the computing of mutual information for normal random variables

If I have 2 correlated normal random variables, and they are not be jointly normally distributed, is there a closed form answer for their mutual information? I've seen that if two normal random ...
1
vote
1answer
27 views

Probability of being in a circle, given normal

Let's assume a bivariate normal distribution with center $\mu$ and covariance matrix $\Sigma$. Let a circle $C$ be given as $C=\{x\in\mathbb{R}^2:||x-\mu||\leq R\}$. I would like to calculate the ...
0
votes
1answer
14 views

Find the limit of the following series of normal random variables.

Let $X_1,X_2,X_3,…$ be a sequence of i.i.d. $N(\mu,1)$ random variables. Then, find $$\lim_{n\to \infty} \frac{\sqrt{\pi}}{2n}\sum_{i=1}^{n}E(|X_i-\mu|).$$ My thoughts: I don't have any rigorous way ...
0
votes
0answers
10 views

Right way to get groups of data based on amount range

I have a database with about 100K records of invoices (date, provider, type and amount). This is sample data: I want to group my data into 4 segments depending on the amount. Group 1: < X1 ...
1
vote
1answer
38 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 ...
-4
votes
0answers
19 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
26 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
16 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} ...
2
votes
1answer
474 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
-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
40 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 $(\bar{X})^2$ != $\mu^2$....
0
votes
1answer
22 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
0answers
42 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)-\...
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 ...
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
1answer
52 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}\...
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 ...
4
votes
2answers
2k views

Bayesian posterior with truncated normal prior

Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal{N}(\mu,\sigma^2)$. The variance $\sigma^2$ is known, $\mu$ isn't. We want to estimate $...
0
votes
1answer
17 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
18 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 ...
1
vote
1answer
2k views

find probability in normal distribution

i would like to check myself if following my answer is correct: let us consider following problem: Suppose the heights of a population of $3,000$ adult penguins are approximately normally distributed ...
0
votes
1answer
526 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also 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 $X \sim N (\mu,\sigma^2)$. ...
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
1answer
30 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
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
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
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
0answers
26 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....
1
vote
0answers
63 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,...
1
vote
1answer
590 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
1answer
410 views

Expected value of cumulative distribution function

Let $\varepsilon$ be a Gaussian distributed random variable with mean $\mu_0$ and standard deviation $\sigma_0$. Is it possible to compute/approximate the expected value $$ \begin{eqnarray} & &...
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 ...
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 ...
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 ...
0
votes
1answer
1k views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and 9....