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

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1answer
24 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
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1answer
13 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
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1answer
37 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
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1answer
20 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
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1answer
12 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
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1answer
13 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
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1answer
15 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
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1answer
23 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
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2answers
42 views

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

How to show the following? $$\large \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2 \pi \sqrt{1-\rho^2}} e^{-\frac{x^2+y^2-2 \rho x y}{2(1-\rho^2)}} dx\ dy=1$$
0
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0answers
19 views

Using Convolution to find density of sum of non-independent normal densities

$X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$. The $X_i's$ are not independent. Let $Y = X_1$ + $X_2$. Then, $ \begin{align*} f_Y(y) &= \int_{0}^{y} ...
0
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1answer
14 views

Probability of Normal Distribution

Let's say that 10 sumo wrestlers were to squeeze into an elevator that could only hold a max capacity of 2300 pounds. Let's say that the weight of the sumo wrestlers is normally distributed with a ...
0
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1answer
19 views

normal probability distribution

If I as just installed 1400 new lightbulbs with an expected mean lifespan of 60 months and a lifespan standard deviation of 10 months. How many bulbs will need to be replaced after 44 months? I ...
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0answers
4 views

Maximum diagonal entry of a multivariate normal sample covariance matrix

Let $\Sigma$ be a covariance matrix of $n$ data points in $\mathbb{R}^p$. So $\Sigma$ is $p\times p$. Suppose that the $n$ points are drawn from the distribution ...
0
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1answer
13 views

Probability of getting an outlier in a normal distribution

Given $ N $ data points that fit a normal distribution, what is the probability that the $ N+1^{th} $ data point is further away from the mean of the distribution than the previous $ N $ data points?
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0answers
19 views

How do I set a problem like this up in my calculator?

"It is known that Lemmings (a small rodent like creature) have a mean body weight of 63.5 grams with a standard deviation of 12.2 grams. If the weights are distributed normally find the probability ...
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2answers
12 views

Normal Distribution problem using the table

the problem goes like this Y has a normal distribution with mean 1 and standart deviation 2. determine P(Y^2 < 9) so i rewrote like this P(Y< sq root 9)=P(P<3)= norm dist ((3-1)/2)=norm dist ...
-1
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1answer
13 views

probability distribution

I would really be grateful if someone could answer me promptly. I believe i should use the poisson distribution model because that is the suitable one however i cannot satisfy the condition of ...
0
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0answers
7 views

How to combine two normal distributions

I want to make a skin detection algorithm based on YCbCr color space. I have a database of $10^7$ triplets (Y,Cb,Cr) which represents a skin color. Now I've computed the normal distribution with ...
1
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3answers
29 views

Multivariate Gaussian, why divide by determinant of covariance matrix?

Given the formula for the density of the multivariate gaussian: $$f_Y(x)=\frac{1}{\sqrt{(2\pi)^n|\boldsymbol\Sigma|}} \exp\left(-\frac{1}{2}({x}-{m})^T{\boldsymbol\Sigma}^{-1}({x}-{m}) \right)$$ Can ...
0
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0answers
12 views

Applying homography to ellipse derived from normal distribution

I need to apply a homography to an elliptic area. First question: Is the resulting also elliptic in every case? I think so, but actually i don't really know. Anyway, I assume it for this question. ...
2
votes
3answers
46 views

Determining $E|X^{n}|$ for $X \sim N(0,1)$ and $n$ odd.

Let $X \sim N(0,1)$. What is $E|X^{n}|$ for $n \in \mathbb{N}$ odd? Attempt: Since $X = -X$ in distribution, we have that $(-X)^{n} = X^{n} = -X^{n}$ in distribution. Then $$E|X^{n}| = E(X^{n})^{+} ...
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0answers
47 views

Normal distribution in nature

I applied for a job as a mathematician. In one of the test questions they asked the following: Why normal distribution is so common in nature? What do you think?
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0answers
7 views

Draw an ellipse corresponding to a bivariate normal distribution

Let $$\mu=\left(\begin{array}{c}\mu_1 \\ \mu_2\end{array}\right)$$ and $$\Sigma=\left(\begin{array}{cc}\Sigma_{1,1} & \Sigma_{1,2} \\ \Sigma_{2,1} & \Sigma_{2,2}\end{array}\right)$$ be ...
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3answers
31 views

Normal distribution notation

I am wondering... is saying $\mathcal{N}\left(0,\begin{bmatrix} 0.1 & 0.02 \\ 0.02 & 0.3 \end{bmatrix}\right)$ equivalent to $\mathcal{N}\left(\begin{bmatrix} 0 & 0 \\ 0 & 0 ...
1
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1answer
15 views

Normal distribution of independent and identically distributed variables

Suppose $X_1,...,X_n$ are independent and identically distributed $N(\mu,\sigma^2)$ random quantities. using the properties of independent normals and expectation and variance operators, explain why ...
1
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1answer
47 views

Sum series of normal pdf's evaluated in normal inverse cdf's

Any hint about how does the following sum grow for k going to infinity? $\sum_{i=1}^{k-1} \phi[\Phi^{-1}(i/k)]$ I "feel" it grows as $k/\sqrt{4\pi}$... but I am not able to prove it. I have also ...
1
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1answer
28 views

How to calculate the Gaussian Integral in specific region?

Firstly, I know that the Gaussian Integral formula, e.g., $\int^{+\infty}_{-\infty}e^{-ax^2}dx=\sqrt{\frac{\pi}{a}}$. But, I am now being encountered a problem when the integral region is not ...
1
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1answer
26 views

Help solving: Normal Distribution problem without using the table OR with a given std

For a recent history test, scores follow the normal distribution with a mean of 70 points. 80% of the students scored below 88 points. What is the standard deviation of the scores? I have done a lot ...
0
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0answers
24 views

Estimate prior for normal distributed data

I have successfully implemented a bayesian classifier using maximum likelihood. In my case I've got 2 classes and I have calculated the two $\mu$ and $\Sigma$. In my problem with a 3-dimensional ...
10
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3answers
134 views
+50

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...
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0answers
11 views

Random Variable of Normal Distribution [duplicate]

Given that a random variable is distributed normally with E(X)=-1 and p(-2<=X<=-1)+p(1<=X<=3)=0.30. Find p(-2<=X<=-1). Please assist me with the steps in solving this problem and ...
0
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0answers
25 views

Normal Distribution of a random variable

Given that a random variable is distributed normally with E(X)=-1 and p(-7<=X<=-2)+p(1<=X<=3)=0.33. Find p(-7<=X<=-2). Please assist me with the steps in solving this problem and ...
-1
votes
1answer
69 views

What is the distribution of $\frac{(X_1 + X_2)^2}{(X_1 - X_2)^2}$ [closed]

If $X_1, X_2 $ is a random sample of size 2 from an $N(0,1)$ population then $\frac{(X_1 + X_2)^2}{(X_1 - X_2)^2} $ follows ?? A) $X^2 _ {(2)}$ B) $F_{2,2}$ C) $F_{2,1}$ D) $F_{1,1}$ Plz help ...
-1
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1answer
43 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
0
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0answers
13 views

Estimate number of data points necessary to generate a normal distribution

I've written a program that generates random normally-distributed variables using the Box-Muller transform. My question is if I can find any formula that relates the number of data points that I ...
2
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0answers
95 views

Finding the limiting probability distribution

I found this problem in Shiryaev's Problems in probability (Problem 3.4.14). Let $\xi_1, \xi_2, \dots$ be a sequence of independent and $N(0, 1)$-distributed random variables. Setting $S_n = ...
0
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2answers
18 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
1
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1answer
54 views

Approximating a Gaussian integral

I have been struggling with an approximation to the following integral \begin{equation} \text{p.v.}\int_{-\infty}^{\infty} {e^{-s^2/2v} \over (e^{-2s}- q a)^2} {ds \over \sqrt{2 \pi v}} \end{equation} ...
1
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2answers
23 views

z score of normal distribution

Good day, I want to ask about standard normal distribution. What is the highest and lowest value of $z$ score can be? From the table of standard normal, the value $z$ score is only for -3.99 $\leq$ ...
1
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1answer
24 views

Calculate the asymptotic dystribution

Let $X_1,...,X_n$ be an i.i.d. random sample from a continuous distribution with density given by: $f(x;\theta)=(\theta-x)\frac{2}{\theta^2}$ if $0<=x<=\theta$ and 0 otherwise. We have the ...
0
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0answers
18 views

Attempted calculation of the probability to win a game.

I'm playing the game "Pepper Panic" and the goal is to create two pepper panics. I noted down some ten results by the numbers I obtained ($0$ or $1$). I obtained a mean of $0.4$ and a standard ...
0
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1answer
13 views

How to find the mean variable of a normal distribution with a given probability and standard deviation?

We have a machine that produces µ g of pasta to be stored in their package, with a standard deviation of 20g. It follows a normal distribution. And we don't want it to produce more than the package's ...
0
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1answer
46 views

how to calculate the marginal distribution of probabilistic principal component analysis

In the book Pattern recognition and machine learning from Bishop equation 12.33 states: $\mathbf{x} = \mathbf{W} \mathbf{z} + \boldsymbol\mu + \boldsymbol\epsilon$ Here $\mathbf{z}$ has a normal ...
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0answers
9 views

How to find value from Gaussian distribution for given point, covariance matrix and expected value.

While reading one article I came across that one of the values (probability) I am supposed to calculate is equal to N(v, b + (h^T)(W^T), I). Where b,v,h are vectors, W is a matrix and I is the ...
0
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1answer
16 views

Let $X_1$ and $X_2$ be independent $n(0,1)$ random variables. Find the pdf of $(X_1-X_2)^2/2$.

I understand that $(X_1-X_2)/\sqrt2)$ ~ $n(0,1)$ since it is a linear combination of $X_1 $ and $X_2$ and hence $(X_1-X_2)^2/2$ ~ $\chi^2_1$. I'm having trouble on how to prove/show this ...
0
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1answer
21 views

How to eliminate coefficients from a sum

For given random values $$X_i \sim\mathcal{N}(0,1)$$ and $$\frac{X_i-\mu}{\sigma}=\tilde{X_i}\sim\mathcal{N}(\mu,\sigma),\,\mu\in\mathbb{R},\,\sigma>0$$ prove ...
1
vote
3answers
64 views

Solving integral related to second moment of normal distribution restricted to interval. CAS answer acceptable

Let $f(x)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$, the pdf of the 1-dimensional normal distribution. Is it possible to compute $\int_{-a}^a x^2 ...
0
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1answer
34 views

How to prove Gaussian integral in normal distribution can be scaled to a standard curve?

If I want to solve the gaussian integral for normal distribution problems I only need to scale it to a standard normal distribution curve and consult a table. I want to know why this is valid (the ...
1
vote
1answer
16 views

Conversion to standard normal

How can I convert a the pdf of a normal distribution that it N(t,1), but integrated from 0 to infinity, to the standard normal. I found that the former is equal to 1- ϕ(-t) but i cant figure how this ...
1
vote
1answer
19 views

Is the joint distribution of two independent, normally distributed random variables also normal?

Say I have $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$, also $X$ and $Y$ are independent, then is the joint distribution of $X$ and $Y$ multivariate normal? I.e., $$\begin{bmatrix} ...