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

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5
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1answer
36 views

Total variation distance of two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$

I need to prove that the total variation distance between two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$ converges to $0$ when $s \nearrow t$. We know that ...
0
votes
1answer
54 views

Estimating the probability of a failure

We are estimating the spares requirement for a radar power supply. The power supply was designed with a mean ($\mu$) life of $6500$ hours. The standard deviation ($\sigma$) determined from testing is ...
1
vote
1answer
47 views

Prove that $Y = \frac{X_1+X_2*X_3}{\sqrt{1+X_1^2}}$ obeys normal distribution

given that $X_1, X_2, X_3$ are independent and identically distributed, $X_1 \sim N(0,1)$. I tried to calculate the cumulative distribution function of Y: \begin{align} P(Y\leq y) &= ...
0
votes
1answer
16 views

sum of iid variables: how many terms needed for convergent to normal

For sum of iid variables $Z_n=\sum_{i=1}^nX_i$, in general, how large should $n$ be to indicate 'convergence' to normal? 10? 100?
0
votes
1answer
30 views

Correlation coefficient and Expectation of two dimensional normal distribution.

Random variable (X,Y) is normally distributed. Conditional expectations are $E(X|Y=y)=0.25y + 2$ $E(Y|X=x)=x-2$ How can i determine correlation coefficient and when that is known, the expectations ...
0
votes
1answer
31 views

Approximation of uniform distribution.

There are leaving from the station arriving every 10 minutes. A person has to wait from 0 to 10 minutes at the station, this is uniformly distributed. Now if the person uses the station 100 times a ...
1
vote
1answer
33 views

finding the pdf for $y$

Let $X\sim N(0,1)$ that is $X$ is a random variable with normal distribution with mean$=0$ and standard deviation$=1$ and $$f_X(x)=\dfrac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$$ Let $y=g(X)=\dfrac{1}{x}$. ...
0
votes
0answers
16 views

Help relating gaussian to chi-squared distribution

I am having trouble finding a simple layout/documentation for the chi squared distribution. From what I understand the chi squared distribution is just: Where "v" is some strength parameter. Now, ...
0
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0answers
9 views

Computing CDF of Normal-Bernoulli

I have the following setting: Let $\theta \sim N(\mu,\sigma^2)$ and \begin{equation} e = \left\{ \begin{array}{l l} E & \quad \text{if} \quad \theta > \theta^* \\ 1 & \quad \text{if} ...
1
vote
3answers
23 views

Characteristic function of a square of normally distributed RV

Let's assume that $ X \sim \mathcal{N}(0,1) $. I'm supposed to compute the characteristic function of $ X^2 $. As far as I got is that the density of $ X^2 $ is $ g(y) = \frac{1}{2\sqrt{2\pi}} ...
1
vote
1answer
34 views

Characteristic function of a product of random variables

I am facing the following problem. Let $X,Y$ be independent random variables with standard normal distribution. Find the characteristic function of a variable $ XY $. I have found some information, ...
0
votes
1answer
49 views

Multiple hypothesis testing

Let's suppose I have 10 independent measurements with results close to zero. How can I claim that they are in agreement with the theory, them being zero? The errors of these 10 results are not equal, ...
0
votes
1answer
14 views

Is the linear transform of joint gaussian necessary gaussian? See this case!

Suppose we map the low dimensional Gaussian distribution into higher dimension using linear transform. Say, $X \in R^p$ is joint Gaussian, and for $n > p$, $Y = A_{n \times p}X$. Is $Y$ joint ...
0
votes
0answers
87 views

Using the central limit theorem to prove a statement regarding normal distribution, from a population with exponential distribution

X1, . . . , Xn are a random sample from a population having an exponential distribution with rate parameter λ. Use the Central Limit Theorem to show that, for large values of n, sqrt(n)*(λx − 1) ∼ ...
1
vote
3answers
79 views

Probability of two normal random variables when random samples are taken from a population

This is sort of second section to my previous question, I should have included both together, but I forgot to. Sorry for any inconvenience. X= random height of a male Y= random height of a female X ...
1
vote
2answers
125 views

Probability of a normal random variable added to a number being greater than another normal random variable, and distribution of average

$X$= random height of a male $Y$= random height of a female $X$ and $Y$ are independent of each other For $x$, $\mu=180\text{ cm}$ and $\sigma^2= 16\text{ cm}^2$ For $y$, $\mu=170\text{ cm}$ and ...
1
vote
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
vote
0answers
28 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
30 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
17 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
votes
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
47 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
votes
1answer
14 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
votes
1answer
26 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
14 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
23 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
28 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
12 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
44 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
7 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
21 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
15 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
21 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
votes
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
votes
1answer
28 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
41 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
33 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
25 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
41 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
9 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
23 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
25 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
19 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
32 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
13 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
1answer
36 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
16 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
12 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 ...