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

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1answer
27 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 ...
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1answer
22 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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1answer
25 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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2answers
13 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 ...
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1answer
22 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 ...
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0answers
156 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 ...
2
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3answers
139 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 ...
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0answers
29 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
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3answers
121 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
79 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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3answers
106 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 ...
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1answer
50 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
69 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 ...
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1answer
52 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 ...
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1answer
36 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 ...
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0answers
30 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 ...
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2answers
466 views

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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1answer
78 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 ...
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0answers
114 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 = ...
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2answers
282 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 < ...
2
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1answer
61 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} ...
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2answers
89 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 $\leq$ ...
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1answer
35 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 ...
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0answers
22 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 ...
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1answer
26 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 ...
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1answer
64 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
35 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 ...
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1answer
18 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 ...
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1answer
28 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 ...
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1answer
168 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 ...
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1answer
194 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 ...
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1answer
25 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 ...
2
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1answer
74 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} ...
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1answer
47 views

Show that two sums have the same distribution?

I have not been able to show that the following two stochastic variables have the same distribution. My question is as follows: Let $$ X_1, X_2,..., X_n $$ be independent and identically ...
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1answer
34 views

Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
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1answer
26 views

Calculating conditional mean of 2 Normal

If $\theta$ is $N(\bar{\theta},\sigma^2_\theta)$, and $s=\theta+\epsilon$, where $\epsilon$ is $N(0, \sigma^2_\epsilon)$, how can I derive that ...
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0answers
24 views

convergence of sequence of functions with finite second moment

Given $0<a<1$. Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ ...
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3answers
89 views

Why do we use only 1/2 for continuity correction in case of approximating binomial random varable to a standard normal random variable?

I have read about continuity correction in case of approximating a binomial random variable to a standard normal variable. But in all the examples , they only use 1/2 as a continuity correction ...
2
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0answers
35 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...
3
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1answer
41 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
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2answers
26 views

Does it matter here that random variables are jointly normally distributed?

My lecture notes ask the following (true/false) question on understanding: Jointly normally distributed random variables are independent iff they are uncorrelated. I don't quite understand what ...
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1answer
24 views

covariance matrix in bivariate distribution

I struggle to understand how exactly you get the covariance matrix in a bivariate normal distribution. The reason is probably that I have no idea how to obtain it at all. In the exercise I have I ...
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1answer
401 views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
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1answer
50 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
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0answers
32 views

Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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1answer
161 views

Dirac Delta function and normal distribution

I understand the Dirac Delta is the limit of a normal distribution when the variance of the normal distribution tends to 0: $$ \delta(x) = \lim_{v\to 0}\frac{e^{-x^2/2v}}{\sqrt{2\pi v}} $$ Then what ...
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2answers
45 views

What is the expected value of a standard normal random variable given value is positive?

Am not sure if I'm wording this correctly. But say we take huge sample of standard normal random variables. Then we separate out positive values. What would be average of the positive values ? What ...
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1answer
18 views

Standardizing Normal Distribution

I was listening to Statistics lecture on Normal distribution and I could not understand that how P(X-mean)/S.D<=(x-mean/S.D) becomes \phi (x- mean/ SD) got solved by chain rule.
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1answer
89 views

Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population.

Recall that for a $N(\mu,\sigma^2)$ population $W=\frac{n-1}{\sigma^2}S^2\sim \chi^2(n-1)$. [a] Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population. ...
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0answers
16 views

How does one scale a covariance matrix learned on de-meaned and scaled data?

I have a dataset on which I want to train a multivariate mixture of gaussians. One common thing to do is de-mean and scale the data such that each feature has zero mean and unit covariance. If I ...