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

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33 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
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1answer
35 views

Mean and variance: Gaussian is the most conservative assumption

"given only the mean and variance of a distribution, the most conservative assumption that can be made about the distribution is that it is a Gaussian having the given mean and variance" I've read ...
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1answer
40 views

stdev and mean from gaussian fit vs. from classical formula

I have a set of data - measured speed of molecules in water. I made a histogram and fitted it with function $$A\exp\frac{(x-B)^2}{C}$$ calculating mean and standard deviation from values B and C If I ...
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0answers
24 views

Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
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17 views

Pivotal quantity that is a function of the z-score: find the CI

** Assumptions ** Let: $X$ be a random variable. $\bar{X}_n$ be the sample mean of X; $\mu$ be the expectation value of X; Assume that $\mu$ is not observable; $S_n^2$ be the sample variance of ...
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1answer
90 views

Statistics: Relationship between process capability and mean

A company produces one-kilogram sugar packets. The specifications on the net content are 1000 ≠ 5 grams. Assuming that the net content follows normal distribution with mean weight as 1005 grams and ...
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0answers
19 views

When $(X, Y)$ are jointly normal given that both $X$ and $Y$ are normal?

We know that $X$, $Y$ are normal does not guarantee $(X, Y)$ is jointly normal. A typical example is: $X=Z$, and $Y=ZU$, where $Z$ standard normal, $P(U=1)=P(U=-1)=1/2$, and $Z, U$ are independent. ...
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41 views

Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
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1answer
134 views

Finding the distribution function of a random variable using CLT

Let $f_0$ and $f_1$ be two continuous probability density functions with means $\mu_0,\mu_1$ and variances $\sigma_0^2,\sigma_1^2$ on $\mathbb{R}$. Furthermore, let $l(y)=f_1(y)/f_0(y)$ be the ...
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1answer
78 views

Bivariate distribution of the sum and product of Gaussian distributed numbers

If $X$ and $Y$ are independent normally distributed random variables $$X,Y\sim\mathcal{N}(0,\sigma^2)$$ How are the sum and product, $X+Y$ and $XY$, co-distributed? You can write the moment ...
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1answer
13 views

Geometric Sequence with Normal Distribution Problem

Given: The running time (in seconds) of an algorithm on a data set is approximately normally distributed with mean 3 and variance 0.25. a. What is the probability that the running time of a run ...
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0answers
6 views

Finding a Gaussian Distribution to approximate a distribution with non-positive definite covariance matrix

We have got a Gaussian distribution covariance matrix(precision matrix) and the potential information, that is, if g is proportional to exp(-X'KX+h'X). However, K here is not positive definite. So we ...
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1answer
20 views

Moments of maximum of bivariate standard normal

Let $X,Y \sim N(0,0,1,1,\rho): f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}}$, and let $Z=max\{X,Y\}$. I'm looking for the first two moments of $Z$. I know it is ...
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2answers
74 views

normal distribution using Z - finding probability between 2 numbers

I am wanting to find the probability of the following: SD = 20 Mean = 100 P(85 < X < 117) i have found the z values for both: P(X>85) : X-u/o = 85-100/20 Z = -0.75 and found the ...
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1answer
42 views
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2answers
26 views

Understanding sampling from a normal distribution with zero mean

I'm studying probability. I came a cross "sampling from distributions". Given a probability density function $f_X(x)$, what I understood is that sampling means getting values of $x$ according to the ...
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1answer
25 views

Bivariate normal distribution when $\rho$ is 0

What happens to the bivariate normal distribution when $\rho$ is 0?The bi-variate normal reduces to a simpler distribution, but what is it? and how do you calculate the cdf then? What I have tried: ...
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1answer
55 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
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3answers
98 views

Gaussian integral evaluation

Asked a question to evaluate the Gaussian Integral, $$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx $$ using the the following approximation, $J=\Bbb E[X^2] \sim J_N = 1/N ...
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2answers
45 views

Normal Distribution Problem

The time taken for a computer to connect to a server is normally distributed with a mean value given by 3.3 seconds and a standard deviation of 0.66 seconds. (a) A computer is said to have a fast ...
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0answers
32 views

Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous?

Let $X$ be a standard Gaussian random variable. Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous ? I don't understand the question here. Now $X$ has density ...
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1answer
52 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$
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0answers
23 views

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
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0answers
15 views

Parameters of normal distribution following other distributions

x follows a normal distribution: x~Normal(μ,σ). However, the two parameters of this normal distribution, μ, σ, follow other distribution. Specifically, μ follows normal distribution: ...
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2answers
54 views

Compute the density of $Y=|X|$

When $X$ has the normal distribution $\mathcal N(\mu,\sigma^2)$ , compute the density of $Y=|X|$ I know ...
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1answer
18 views

using standard normal deviation to calculate mean?

if i have an unknown mean, a standard deviation of 4, and P(X < 8 ) = 0.3085, how do I calculate the mean somehow using the standard normal distribution and it's cummulative function? I know that ...
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0answers
60 views

Finding the Probability Limit and Asymptotic Distribution of Xbar-LogYbar

I'm kinda still new to Large Sample Theory and I have already attempted the question. Not sure if I did it right. Based on Kinchin , I know Xbar converges in probability to mu and Ybar converges in ...
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0answers
29 views

Let Xi be iid with EXi = mu and Vxi = sigma^2. Find the asymptotic distribution of Xbar^2

I don't know why I'm having so much trouble with this question. I am supposed to do it in two ways and the first way was using the delta method. And I hope I did it right. However the question ...
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1answer
36 views

The number of coin tosses needed if the proportion of heads is to lie within 0.05 of p with probability at least 0.9?

There's a question I'm not really sure if I did it right or even understand what its trying to say. There is a coin which produces heads with an unknown probability p. How many times should we throw ...
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1answer
25 views

Probability Distribution of z/x given x

It may seem a simple question for you, but it's driving me crazy. Given the regression model $z = wx + \epsilon$, where $ \epsilon \sim \mathcal{N} (0, (\sigma x)^{2} $, $ z \sim \mathcal{N}(wx, ...
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0answers
8 views

Performance of an optimum estimator for Gaussian random variables used against Non-Gaussian random variables

Consider an optimum estimator for some parameter where the underlying distribution is following a Gaussian distribution with mean 'mu' and standard deviation 'sigma' (denoted by N(mu, sigma)). Let ...
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1answer
47 views

Normal distribution percentile calculation

I'm working out the following problem and there is a part that I am not understanding clearly. The weight distribution of parcels sent is normal with mean value $12$ lbs and standard deviation ...
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1answer
45 views

derive the mean and variance of $\bar X$ using means of sums rules

I can't find anywhere what the means of sums rules are so i'm confused with this question The random variables $X_1......X_5$ are jointly multivariate normal. Their expectations are $E(x)= \mu_i$ and ...
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0answers
24 views

Multivariate Normal Variables

Sorry if this is a bit hard to read. Not entirely sure how to use the MathJax formatting on the site... The random variables $X_1, X_2, X_3, X_4$ and $X_5$ are jointly multivariate normal. Their ...
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1answer
183 views

Mill's Inequality on normal distribution

Given that $Z \sim N(0,1)$. Prove Mill's Inequality: $$P(|Z|>t) \leq \sqrt{\frac{2}{\pi}}\frac{e^ {\frac{-t^2}{2}}}{t} ~\forall t > 0$$
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1answer
34 views

To calculate $E(Y|X)$ and Var$(Y|X)$.

Suppose $U $ and $V$ are independent and each is distributed as $ N(0,1$). Define $ X$ and $Y$ by $Y=X-1-U$,$ X=2Y-3-V$ . What is $E(Y|X)$ and Var$(Y|X)$ ? Again another questions which I'm unsure ...
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0answers
73 views

Find the best predictor and the best linear predictor of $Y^2$ given $X$. Suppose $(X, Y ) \sim N(0, 0, 1, 1, p ).$

Once more, there's another question that I'm clueless on how to start. I should have dropped this course earlier. Suppose $(X, Y ) \sim BN(0, 0, 1, 1, p )$, meaning that $X$ and $Y$ are bivariate ...
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0answers
22 views

Antithetic pair of non-independent normal random variables

Suppose that I have two non-independent normal random variables, X and Y such that $(X,Y)$ has mean 0 and the following variance covariance matrix: \begin{bmatrix} 1 & \rho ...
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1answer
25 views

Calculate a probability involving drawings from bivariate normal variables with Xi and Yi i.i.d

There's a question which has been troubling me along with my earlier post. To be honest, I'm not entirely sure on how to proceed. All I know is that if X~N(mu,sigma^2) then P(X < A) = P(Z< ...
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0answers
27 views

Conditional covariance in gaussian graphical models

I have a hypothesis, but I'm not sure if its true. The Wikipedia page states that if the covariance matrix is given by $$\Sigma=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right]$$ ...
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1answer
20 views

Calculating number on normal distribution curve

Can someone please let me know if I have this question correct: ...
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1answer
33 views

How to calculate the value of $E[X^4], E[X^6],E[X^8] $…?

I learned that when X is a normal random variable , $X$~ $N(0,1)$ , $E[X^2]=1$ $E[X^4]=1.3=3$ $E[X^6]=1.3.5=15$ $E[X^8]=1.3.5.7=105$ For the general case , when variance is s , how do you do ...
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0answers
46 views

Why are the real part and imaginary part of normal distribution function independent?

As I said in title, why are the real part and imaginary part of normal distribution function independent? I need a detail derivation to proof it. Thank you.
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1answer
25 views

Normal distribution probability function definition

Up to now, I believed that k-dimensional normal distribution has probability function: $\frac{1}{\sqrt{(2 \pi)^k |\Sigma|}}e^{-\frac{(x-\mu)^T\Sigma^{-1}(x-\mu)}{2}}$ Recently I have read an article ...
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0answers
42 views

Model selection: geometric mean of the standard deviation.

I have two models that represent a physical process. To determine which model is the best, I make some experiments and compare measured data with data predicted by each of the models. The model with ...
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1answer
38 views

Where are they getting this number from?

Here's the question that I'm having a problem with: ...
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1answer
26 views

CLT approximation

Let $X_1,\ldots,X_{735},Y_1,\ldots,Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$ and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Find $P(\sum_{i=1}^{735} X_i ...
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1answer
15 views

Sum of variation for loads

The loads on an electrical network with 10 regions are modelled by considering a base load with mean 20mW and standard deviation 3mW. Variation due to regional load is modelled by considering that ...
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0answers
13 views

Sum of random variables for 2m tape

we use 2 metre tape for distance measurement and that the measurement error for the full tape length has 0 mean and variance 1.5cm^2. Find the mean and the variance if the total distance measured by ...