Tagged Questions

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

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1answer
12 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 ...
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2answers
16 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? ...
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0answers
11 views

Statistic z scores [on hold]

. 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 ...
2
votes
1answer
16 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
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0answers
17 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
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0answers
31 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
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0answers
27 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
11 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 $\|P_VX\|^2$ ...
0
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0answers
4 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
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0answers
10 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
23 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
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0answers
30 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
16 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
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1answer
24 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
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1answer
13 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
28 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
14 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
43 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
20 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
14 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
16 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
24 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
46 views

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

How to show the following? $$\large \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2 \pi \sqrt{1-\rho^2}} e^{-\frac{x^2+y^2-2 \rho x y}{2(1-\rho^2)}} dx\ dy=1$$
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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
votes
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
20 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
21 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 ...
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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
32 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
13 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
52 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
32 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
vote
1answer
18 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
vote
1answer
49 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
vote
1answer
27 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
26 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 ...
12
votes
3answers
192 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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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 ...
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votes
1answer
70 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 ...
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votes
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 ...