Tagged Questions

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
9 views

The effect of a decrease in sample size on a confidence interval

I have a some data that contains 100 elements. I can model it as a normal distribution using the t-distributionI have used the t-distribution to construct a confidence interval for unknown value of ...
1
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0answers
16 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 ...
1
vote
1answer
29 views

Determine probability of fewer than a certain number of events

Could anyone help with the following problem? My guts is telling me that the answer to part (a) is a normal distribution. Mainly, because I can't see where a uniform distribution would fit in this ...
1
vote
1answer
16 views

Calculating probability using normal tables

I've had a crack at this question however I don't seem to be getting the correct answer and I can't figure out why. I've been given a table of the 'Normal Distribution Function' where the left tail is ...
0
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0answers
12 views

Confidence interval issue

n (sample size) = 96 Average value x¯ = 4.1 σ = 4.5 I have to calculate the confidence interval's lower endpoint a and upper endpoint b for expected value 99,5%. (Normal distributed). So far I ...
0
votes
1answer
41 views

Find the distribution of sum of random variables given bivariate distribution.

$\bullet$If $(X1, X2)$ be a bivariate Gaussian random variable with parameters $µ$ and $Σ_{x}$. Find the distribution of $X1 + X2$. Hi all, for this question I'm not sure about the best way ...
0
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0answers
9 views

Solve for mean and std deviation of new normal distribution

There are normal distributions with known means and standard deviations. The first distribution is a Bayesian prior distribution with known mean1 and known SD1. The second distribution is a Bayesian ...
0
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0answers
10 views

Bayesian Uni-variable ou multi-variable and formulation

I have a parameter that has a prior distribution with mean equals to 30, a variance of 25 and a number of samples $n=30$. I was able to execute 30 more samples, and I got a mean of 25 and a variance ...
0
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1answer
19 views

What algorithm to solve this integral similar to a normal CDF numerically?

I'm looking to solve this integral numerically. It is a bit similar to a normal CDF. z and tau are deterministic. What kind of algorithm may do the job, ideally to be coded in C/C++? I'm ...
2
votes
1answer
13 views

Covariance of a mixture of Gaussians

I have seen this question asked, but in a strange way that I do not think is equivalent. If someone can show that the formulations are identical, I would be grateful. Suppose with probability $p$, ...
0
votes
1answer
20 views

Median of truncated / limited normal distribution

The peoples weight is normally distributed $\mathcal{N}(0,\,1)$ The $\mu \; , \; \sigma \; and \; \sigma^2$ are known. How can i calculate the median weight of people if everyone who weights less ...
0
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0answers
23 views

normal distribution questions

In a maths class, students are given a maths question. The mean time it takes them to complete this question is 2 minutes and the standard deviation is 30 seconds. In your answers, you may assume ...
1
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1answer
11 views

Find percentage of conforming items for normal and uniform distributions

I'm given the following problem: Could anyone give some insight into how to solve this? I understand that for the normal distribution, I will likely have to look the percentage up in a table. ...
4
votes
1answer
39 views

Normal distribution - how to solve P(-b<X<b)=0.95

$X\sim N(2,3^2)$ How do you find $b$ where $P(-b<X<b)=0.95$ other than trial and error? You can't directly transform to $z$ because if you find an appropriate $z$, transforming back will give ...
0
votes
0answers
6 views

Is there a measure that monotonically changes as we change a normal distribution to a bimodal distribution?

Is there a measure with values varying monotonously from -1 to 1 (or 0 to n) as we change a bimodal distribution (or say a reverse normal distribution) to a normal distribution? Will the same work if ...
0
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0answers
19 views

Normal distribution: how to approach a simple problem.

how could I compute this probability? $$P(|X-2|>5)\text{ if }X\stackrel{d}{=}\mathcal{N}(1,4).$$ $|X-2|>5\Rightarrow 7<X<-3\Rightarrow 3<Z<-2$. But calculating the probability to ...
1
vote
0answers
12 views

Show the diffusion equation is a normalised distribution.

The diffusion equation is defined to be $$P(x,t) = \dfrac{1}{\sqrt{4D\pi t}} \exp \left(-\dfrac{x^2}{4Dt}\right),$$ where $D$ is a physical constant. Show that the reaction diffusion equation is a ...
0
votes
1answer
20 views

Inverse SNR: find the first point with a specified SNR ratio where noise and signal are simple normal distributions

I have a pair of 2 simple normal distributions for noise and signal , specified by $\mu1,\sigma1$ and $\mu2,\sigma2$, so I know how to calculate CDF1, CDF2 for every point. I would like to find $x$ = ...
0
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0answers
7 views

Any percentile formulas for multivariate Gaussian?

I am working on a classification problem. For 1-d Gaussian distribution, it is well known that 99.7% of the values are not more than 3 sigma away from the mean. Thus if one is developing a binary ...
0
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1answer
27 views

Finding the variance of speeds

This is a question from my Statistics textbook which I am currently stuck on. I have approached the question in a couple ways but each time I have been incorrect. A summary of the speeds, x ...
0
votes
1answer
40 views

How to find expected value of a portion of the normal distribution?

$X\sim N(67,4)$ What's the expected value of the portion of the curve $(X>72)$? I tried to use the definition of expected value ($\int xf(x) \mathrm{d}x$), but my integral was far too complicated ...
3
votes
1answer
32 views

$E[X_1^2X_2^2]$ for a gaussian vector

I would like to calculate the following expectation $E[X_1^2X_2^2]$, where $X=(X_1,X_2)$ ha a gaussian distribution with mean vector $(0,0)$ and covariance matrix Q (non diadonal and non invertible). ...
0
votes
0answers
24 views

Working out percentage from Normal Distribution

I'm having an issue with this question and was hoping someone could point me in the right direction A sample of $200$ screws produced by a machine has a mean weight of $5.02g$ and a standard ...
2
votes
1answer
34 views

What is the joint distribution of sample mean and sample variance of normal distribution?

${X_i} \sim N\left( {\mu ,{\sigma ^2}} \right)$, define $\overline X =\dfrac{1}{{n}} \sum\limits_{i = 1}^n {{X_i}} $, ${S^2} = \dfrac{1}{{n - 1}}\sum\limits_{n = 1}^n {{{\left( {{X_i} - \overline X} ...
0
votes
0answers
6 views

How to determine how likely it is for a data point to belong to each of two [normal] distributions?

How do I determine the likelihood of a data point to belong to a distribution? Context I have a set of data, which has the following histogram: I would like to analyze each of the data points in ...
0
votes
1answer
20 views

Distribution under null-hypothesis and type 1 error

Given random variables $X_1,...,X_n \overset{i.i.d.}{\sim} N(\mu, \sigma^2)$ where the variance $\sigma^2$ is known let the null hypothesis be $H_0: \mu = \mu_0$ For the statistic $T=\sum_{i=0}^nX_i$ ...
0
votes
1answer
19 views

statistic normal distribution

Transport Canada was investigating accident records to find out how far from their residence people were 2 when they got into a traffic accident. They took the population of accident records from ...
1
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2answers
22 views

statistics z score

I'm curious what the phrase "on average" means. Here is an example: On average, 30% were further than ___ kilometers away when they had their accident. Is $30\%$ a $z$-score or is it a mean? ...
2
votes
1answer
17 views

Normal distributions sums

I read this property about normal distribution If $X\sim\mathcal N(\mu_X,\sigma_X^2)$ and $Y\sim\mathcal N(\mu_Y,\sigma_Y^2)$ are independent, then $$ X+Y\sim\mathcal ...
0
votes
0answers
29 views

Asymptotic expansion at infinity of integral function

Given $q\in(0,1)$ find $z$ such that $$ F(z)\equiv\int_{-\infty}^{z}\frac{e^{-\frac{y^2}{2 \sigma _{22}^2}} \text{erfc}\left(\frac{\rho \sigma _{11} y-\sigma _{22} V}{\sqrt{2-2 \rho ^2} \sigma ...
2
votes
1answer
12 views

How is this Variance found in this old question?

On this question: Probability: Normal Distribution they find these values: $\hat\mu = .05(150) = 7.5\space,\hat\sigma = \sqrt{150(.05)(.95)} = 2.67$ I see how they got $\mu$, but how did they get ...
-2
votes
0answers
45 views

Gaussian Random vector with zero mean

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
2
votes
1answer
34 views

CI for the expected value of the sum of two dependent normal RVs

Let's consider 2 dependent, normally distributed R.V.s, $X_1$ and $X_2$. The means, $\mu_1$ and $\mu_2$ are known, as well the covariance matrix $\Sigma$. Let's consider the following random ...
2
votes
1answer
15 views

distribution of $\|P_VX\|^2$ with orthogonal projection $P$ onto $V$

We've had the following question discussed today but without any result: Let $X_1,\dots,X_d$ be random variables, iid and $X_n\sim N(\mu_n,1)$. How can we describe the distribution of $\|P_VX\|^2$ ...
1
vote
1answer
10 views

Gaussian conditional distribution

Let $Y_t$ be a gaussian process with $E[Y_t]=0$ and $Z=\frac{\int_0^1 Y_s ds}{\sqrt{V}}$ where $V=Var(\int_0^1 Y_s ds)$ (so Z has a standard normal distribution). I want to prove that conditionally on ...
0
votes
0answers
11 views

Convergence in distribution of a serie

How could we prove that this serie converge in distribution to a centered gaussian variable ? $$ \frac{1}{\sqrt{n^3}} \sum_{i,j,k = 1}^{n} x_{i,j} x_{j,k} x_{k,i} $$ with for all $ i,j \in ...
3
votes
1answer
29 views

find the moment generating function of a pdf

Let $X$ be a random variable with pdf $$f_x(x)=\frac{1}{2\sigma}e^\dfrac{-|x-\mu|}{\sigma}$$, $-\infty< x<\infty$, $-\infty< \mu<\infty$, and $\sigma>0$. I have to find the mgf of $X$?. ...
0
votes
0answers
33 views

Covariance of two normally distributed random variables

$$X = \frac1N \sum_1^N d_icos\theta_i $$ $$Y = \frac1N \sum_1^N d_isin\theta_i $$ $$d_i \sim LN(m_i, \sigma^2) $$ (LN mean Log normal distribution, $m_i$ can be measured, actually function of ...
1
vote
1answer
18 views

Distribution of orthogonal projection onto $\{(x,\dots,x\}\subset \mathbb R^d$

Let $D:=\{(x,\dots,x)\mid x\in\mathbb R\}\subset\mathbb R^d$ and $X$ be a random variable and normally distributed $X\sim N(\mu,\sigma^2 E_d) $ with $\sigma\neq 0$, $\mu\in\mathbb R^d$ and $E_d$ the ...
0
votes
1answer
25 views

Normal approximations and Binomial distributions

I am having some difficulty with the following question from my textbook. I have really been trying to understand the use of normal and binomial approximations, but I'm getting really confused. Any ...
0
votes
1answer
17 views

Probability Between Two Normally Distributed Variables

The weight of medium-size tomatoes selected at random from a bin at the local supermarket is a normal random variable with mean $μ = 10$ ounces and standard deviation $σ = 1$ ounce. Suppose we pick ...
0
votes
1answer
29 views

Normal Distribution $E(X^4)$?

So I have the Normal Distribution $f(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}.$ I know any $E(Z^{\mbox{ (any odd #)}})$ makes you integrate an odd function thus giving an answer of zero (i.e. $E(Z^1)$ and ...
0
votes
1answer
15 views

Normal Distribution finding the probability of having enough is 95%

Suppose that ice cream consumption per person at parties is normally distributed with a mean of 0.39 gallons, and a standard deviation of 0.26 gallons. If you are throwing a party with 33 guests, how ...
0
votes
1answer
50 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
0
votes
1answer
21 views

Normal distribution tail probability inequality

I am trying to show that $$P(X>t)\leq \frac{1}{2}e^\frac{-t^2}{2}$$ for $t>0$ where $X$ is a standard normal random variable. Perhaps this is simple. I have been starting with $$ ...
1
vote
1answer
23 views

Normal distribution and conditional probability in $\Bbb R$

Normal distribution with a mean of $28.3$ and a standard deviation of $0.77$. We know that $X$ is at least $27$, what is the probability that $X$ will be between $29$ and $40$. I have calculated ...
1
vote
1answer
16 views

Probability that two points (any where on the curve) are a set number of standard deviations apart on a normal distribution

So, here is the question: You buy two pieces of pipe from supplier A, and the inner diameter has a normal distribution of N(muA, sigmaA^2) = N(8.02, 0.1^2). You want these two pipes to butt together ...
1
vote
1answer
17 views

How to find a probility that the sample mean of a population lies in a particular range?

Assume that X is a random variable with mean x_mean and standard deviation x_sd. If we take a sample of ...
0
votes
1answer
29 views

Expectation of a powered complex circular gaussian process

Assuming a complex circular zero-mean gaussian random process (or vector) $\textbf{x}$ $\left(\textbf{x}\sim \mathcal{CN}\left(0,\sigma^2\right)\right)$. $\mathbb{E}\{\textbf{x}\}=0$. The question ...
2
votes
2answers
74 views

How to show that the integral of bivariate normal density function is 1?

How to show the following? $$\large \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2 \pi \sqrt{1-\rho^2}} e^{-\frac{x^2+y^2-2 \rho x y}{2(1-\rho^2)}} dx\ dy=1$$