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

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-1
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1answer
40 views

Normal distribution with dice

I'm wondering how to control the normal distribution that comes from summing dice rolls only using different numbers of dice, different combination of types of dice (d4, d6, d8, d10, d12, d20) and ...
2
votes
1answer
25 views

Expectation of product of two correlated gaussian variables

$\newcommand{\var}{\operatorname{var}}$It seems I can not find the answer anywhere, please point it out how to calculate. Here, I have $X$, $Y$,$G$,$X_D$ and $Y_D$,both are Gaussian variables, and ...
0
votes
1answer
35 views

Variance of a Gaussian Random Variable

Show Variance of a Gaussian random variable $N(\mu,\sigma^2)$ and I know $\mathbb{E}(X)^2 = \mu^2$. So I need $\mathbb{E}(X^2)$ = $\int_{\mathbb{R}} x^2 \frac{1}{\sqrt{2\pi\sigma^2}} ...
0
votes
1answer
34 views

Sigmoid function that approaches infinity as x approaches infinity.

The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity. The ...
0
votes
0answers
12 views

Regarding the distribution of pivotal functions not depending on their parameter(s)

I have difficulties understanding the part of pivotal functions not depending on their underlying parameters. Let's take a simple example, if Y is a random sample from an $N(\mu,1)$ distribution and ...
2
votes
0answers
47 views

Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
1
vote
1answer
23 views

finding variance of gaussian distribution from mean

The Gaussian random variable $X$ can be used to model the number of customers that enter a market in 1 minute at a given time of the day. The mean number of customers that enter the market in 1 minute ...
1
vote
3answers
39 views

Finding the probability of loss from standard deviation in normal distribution

I am unsure how to approach the following question. The returns from a project are normally distributed with a mean of \$220,000 and a standard deviation of \$160,000. If the project loses more than ...
0
votes
1answer
18 views

Representation of a non-standard normal variable squared

I have come across a representation of a non-standard normal distributed variable square. It is clear for me that assuming $Z_j \approx N\left ( \theta_j, \frac{\sigma^2}{n} \right )$ we can write ...
0
votes
1answer
27 views

normal approximation of binomial distribution

a school buys 60% of its light bulbs from supplier A and 40% from supplier B. the light bulbs from both suppliers look identical but light bulbs from supplier A have exponentially distributed ...
1
vote
2answers
34 views

Gaussian distribution raised to a power

Given that $X$ follows a Gaussian distribution $e^{-x^2/2\sigma^2}$, what distribution is followed by $X^{1/3}$? How does one start to solve this problem? I guess it isn't ...
0
votes
0answers
23 views

Inequalities in binomial and normal distrubutions

Example Q Foo is normall distrubuted like $$X\sim N(100,15^2)$$ foo of 110 is required. Does that mean that I should find: $$P(X\gt 109) $$ or $$P(X\gt 110) $$ or $$P(X\ge 110) $$ I feel ...
0
votes
1answer
19 views

How to Normalize the Sum of Two Gaussians

I have the following function: $I(\theta_i) = I_0 + I_1\exp(\mu(\cos(\theta_i - \theta_s) - 1))$. Suppose I have two implementations of this function, whose parameters match with the exception of ...
2
votes
0answers
27 views

Characteristics function and moments of multivariates

I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf This is a paper by Nengjiu Ju who uses talyor series to express ...
0
votes
1answer
25 views

Calculate the Probability of a Normally Distributed Random Sample

Please i would like to understand these problems about probability distributions, I can't find a right solution for this problem. I have a variable X which is the level of glucose in blood and is ...
0
votes
0answers
54 views

expected value minimum of bivariate normal distribution

Let $X,Y$ be jointly normal with density $f(x,y)=\dfrac{1}{2\pi\sqrt{1-\rho^2}}\exp(-\dfrac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2))$. Let $Z=\min(X,Y)$. Show that $E[Z]=\sqrt{\dfrac{1-\rho}{\pi}}$ and ...
1
vote
1answer
18 views

Understanding the normalization of a Gaussian

I have a Gaussian defined as follows: $W(\theta) = j * exp(-0.5 * \theta^2 / \sigma^2)$. I want to set $j$ such that $\frac{1}{360}\int_{-180}^{180} W(\theta)d\theta = 1$. I'm using two values for ...
0
votes
1answer
22 views

variance of multivariate normal

currently trying to compute the first two moments of the multivariate distribution. Got an extremely helpful answer to show that $\mathbb{E}[x]=\mathbb{m}$, with $x \sim ...
1
vote
1answer
26 views

Would the joint distribution of Normal Random Variable and the distribution of a X bar from the same sample be bivariate Normal?

I know this question is somewhat redundant... but here goes: My text asserts that the joint distribution of $$X_1=N(\theta, 1)\text{ and } \bar X = N(\theta, \frac 1n)$$ is Bivariate normal with ...
1
vote
1answer
74 views

The probability that the ratio of two independent standard normal variables is less than $1$

Let the independent random variables $X,Y\sim N(0,1)$. Prove that $P(X/Y < 1) = 3/4. $ Could anyone help me prove this analytically? Thanks. Progress: My first thought was to integrate the joint ...
2
votes
1answer
39 views

Normal Distribution Probability with known mean and variance

I believe I am quite close to solving this, but I would just like to double check some of these answers. Two species have different size toes. Lengths of toes of species X is normal distributed with ...
3
votes
1answer
24 views

Basically Normal Dist question

I'm a little rusty on my probability and would appreciate any help. I think I have done the bulk of the work already anyway, but my question is: If $X \sim LN(1,2)$ find $P(X>1)$ $X$ being ...
0
votes
1answer
24 views

multivariate normal moment derivation

I am having trouble deriving the mean for a multivariate normal for $\mathbf{x} \sim \mathbb{N}(\mathbf{m},\Sigma)$: $$ \mathbb{E}[\mathbf{x}]= \int_{R^d} \mathbf{x} ...
4
votes
1answer
109 views

Bayesian Updating with 1 Signal but 2 Unknowns

Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 different values {${\alpha_1, \alpha_2}$} such that $\alpha = \alpha_1$ with probability $p_1$ and $\beta$ is ...
0
votes
1answer
29 views

Distribution with density $x^2\operatorname{exp}\{-x^2/2\}$

I came across the probability distribution with density $$ f(x)=\sqrt{\frac{2}{\pi}}\,x^2\,\mathrm{e}^{-\frac{x^2}{2}},\quad x\geqslant 0. $$ Is this distribution known under a certain name? I only ...
0
votes
0answers
14 views

The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution

Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ...
2
votes
1answer
76 views

Simple calculations of mean, standard deviation, and probability

You are a successful entrepreneur that has developed a new sustainable product that is manufactured through a standard production process. As part of this process, the product goes through quality ...
0
votes
2answers
34 views

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve? a) $P(0 < Z < 2.17)$ b) $P(-2.50 < Z < ...
1
vote
0answers
16 views

Normal distribution tables - right or left?

Are the probabilities in normal distribution tables given typically to the right or left of the $Z$ score? One such text I am reading says to the right. However, in my lecturer's exercises, I ...
0
votes
0answers
17 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
0
votes
1answer
45 views

Generate a uniform distribution from n coin flips

I'm making a computer game and I've reduced the problem into something simple: How can I show the player the number of heads he "tossed" given some number of coins = n? Naive expected value is ...
0
votes
0answers
22 views

Moments of Multivariate Normal Distributions

I have two questions. Suppose we have two multivariate normal distributions $X \sim N(\mu,\Sigma)$ and $Y\sim N(c\mu,\Sigma)$ where $0<c<1$ is a constant, $\mu$ is a vector and $\Sigma$ is a ...
2
votes
1answer
29 views

Find probability given a binomial and a normal distribution

$X$~$Bin(n,p),Y_n$~$N(μ,\sigma^2)$ Where X is the number of trials taking place, and $Y_n$ is the amount of time the $n$th trial takes (independent of other trials). $Z$ is a new random variable ...
1
vote
2answers
18 views

Calculating a normal distribution with a sample size?

the sample of $n=25$ is what is throwing me off. I have no clue what to do with it. Given a normal distribution with $\mu=101, \sigma=25$, and given you select of $n=25$ $A.)$ $P(\overline{X} ...
1
vote
0answers
24 views

Approximating the integral of the exponential of a quadratic function

An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where $c=b^2/(4a)+\ln(-a/\pi)/2$ and ...
0
votes
0answers
29 views

How to construct a two sided confidence interval?

A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected and the diameter is measured. The resulting data are shown below. 5.21 5.28 5.29 5.27 ...
1
vote
0answers
18 views

Finding the conditional distribution of a normal RV given another normal RV

I'm having trouble with this question from a past qualifying exam: Question Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. ...
1
vote
0answers
25 views

Can a mixture of normals be a constant?

Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane?     ...
0
votes
0answers
23 views

Expectation of Multivariate Normal - help please

I am struggling to show why an equation is true. Help would be very much appreciated Given that x $\sim$ N($\textbf{m}$, $\Sigma$) [multivariate normal], show that: E[($\textbf{x}$ - ...
0
votes
1answer
27 views

Square Matrix Algebra - help please!

I am stuck on a problem in matrix algebra and I would be happy if someone could help me. Given a square matrix with dimensions "p" given that $\textbf{x}$ $\sim$ N($\mu,\Sigma$) [multivariate ...
1
vote
1answer
40 views

How to interpret the covariance matrix of Brownian motion

I'm reading Bernt Oksendal's "Stochastic Differential Equations". It says, Brownian motion $B_t$ is Gaussian Process, i.e. for all $0 \leq t1 \leq \cdots \leq t_k$ the random variable $Z = ...
1
vote
0answers
45 views

If $X|Y$ and $Y$ are both normal, is $X|Y>y$ normal as well?

Consider two random variables, $X$ and $Y$, with the following properties: $X|Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X|Y>y$ follow a normal distribution as well? If so, what are its ...
5
votes
2answers
33 views

correlation between $\sum_{i=1}^{98}X_i$ and $\sum_{i=3}^{100}X_i$

Let $X_1,...,X_{100}$ be iid $N(0,1)$ random variables. The correlation between $\sum\limits_{i=1}^{98}X_i$ and $\sum\limits_{i=3}^{100}X_i$ is equal to (A) $0$ (B) $\dfrac{96}{98}$ (C) ...
0
votes
1answer
50 views

Central Limit Theorem sample vs population

I need help in the setup of this problem. I'm sure that I'm making this far more complicated than what it actually needs to be. "An anthropologist wishes to estimate the average height of men for a ...
0
votes
0answers
18 views

If I approximate a Bionimial distribution with a Normal Distribution am I still allowed to use Binomial's equation for Variance?

If I approximate a Binomial distribution with a Normal Distribution am I still allowed to use Binomial 's equation for Variance? So am I still allowed to use this: $Var(x) = np(1 − p)$ While still ...
0
votes
0answers
55 views

Expected value of a function of truncated normal

I need to find the expected value of the following type of an expression: $$\mathbb{E}[\frac{1-\alpha}{1-\alpha-\frac{X}{\beta}}]$$ where $\alpha$ and $\beta$ are constants and $X$ is a random ...
1
vote
0answers
35 views

Question Relating to the Central Limit Theorem

I have the following question: Suppose $X_1,X_2, \ldots, X_{12}$ are identical independent uniform random variables on $[0,1]$. Let the sample mean(lets call it $m$) = $\frac{1}{12}(X_1 + X_2 + ...
0
votes
0answers
18 views

Distribution of unknown, given system of equations

Suppose we have an unknown real $x$. We want to give an approximation of $x$ by measuring the distance between $s_i$ and $x$, for $i = 1,2,3$. The position of each $s_i$ is distributed with mean $p_i$ ...
0
votes
1answer
22 views

Calculate $E(X^2)$ of random variable $X$ ~ $N(3,4)$

I need to find $E(X^2)$ of random variable $X$ ~ $N(3,4)$. I can use the simple way: $E(X^2) = \int_{-\infty}^{\infty} x^2 \cdot f(x) dx$, in this case $f(x) = normal \space distribution ...
0
votes
0answers
20 views

Poisson distribution with normal informative priors

I'm Jia, a student of economics and finance. I was wondering if someone could help in understanding this problem. I've just started to attend a new course "Financial and nonlinear econometrics" and ...