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

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0
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2answers
11 views

Normal distribution with standard deviation = I

Suppose a vector $\epsilon \in \mathbb R^d$ is a random vector drawn from the isotropic normal distribution: $\epsilon$ ~ $\mathcal N (0, I)$ [As in Eq. 1.34 here.] I suppose ...
0
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0answers
8 views

How does the formula for standard deviation result in the normal distribution

Trying to understand this is in a high school level. I understand that the how $\frac {\Sigma|x-\bar x|}{n}$ calculates the mean of the distances of each score to the mean. I use this idea to map ...
0
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0answers
7 views

How to calculate distribution of (X1, X2) conditional on (C1, C2)?

Say that $X_{1}$ = $a_{1}$$X_{2}$ + $B_{1}$$C_{1}$ + $E_{1}$  , and         $X_{2}$ = $a_{2}$$X_{1}$ + $B_{2}$$C_{2}$ + $E_{2}$  , ...
0
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0answers
11 views

SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates ...
0
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0answers
29 views

Solving for the limit of a Gaussian random variable within an integral

I'm having trouble solving a particular integral. It is $$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$ where $$ I(t') = \mu_c+\sigma_c \eta(t'). $$ In this second equation, $$ \eta(t') = ...
1
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2answers
25 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
0
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2answers
23 views

Generating points from 2 Normal distributions and $0$-probability continuous r.v.s

Consider the following experiment: We generate "green" points and "blue" points in $\mathbf{R}$ using two different normal distributions as follows: 1000 green points are sampled from a $N(-1, 1)$ ...
-4
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0answers
22 views

Expected lifetime [on hold]

I have 5 machines (elevators) and would like to know "a realistic worst case" in terms of when they break down and need to be replaced (for budgetting). They run independently and so far for 62 years ...
2
votes
1answer
30 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
2
votes
1answer
47 views

Gaussian function in the limit of trigonometric functions

I've noticed that $$ (\sin\theta \cos\phi)^{2n} + (\sin\theta \cos\phi)^{2n-1} $$ increasingly resembles a Gaussian function of $(\theta, \phi)$ as $n$ goes to infinity. In particular, when I take ...
0
votes
1answer
40 views

If $X$ and $Y$ are Normally distributed with correlation $\rho$, can we say anything about $E[Y \mid X]?$

Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$? In general, this clearly does not seem to work, ...
6
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0answers
55 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is ...
-1
votes
1answer
28 views

expectation of product of sums of normally distr. r.v.

Let $Z_1$ and $Z_2$ be i.i.d. standard normally distributed. $X_1=Z_1+Z_2$ and $X_2=Z_1-Z_2$. Apparantly E[|$X_1|*|X_2|$] = E$[|Z_1|*|Z_2|]$. Why?
0
votes
1answer
27 views

What is the PDF, CDF, and E[Y] of Y=ln[X+c] if X is lognormal

If $\ln X \sim N(\mu, \sigma^2)$, what is the distribution of $Y=\ln \left(X+c\right)$ where $c$ is a constant. Is this something that can be written out analytically? Also, what is $E[Y]$?
0
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0answers
31 views

Distribution or samples of a function of a random variable

OK I edited the question: I have the following setup: Stereo camera setup with two images I, I'. 4 1-dimensional random variables (each corresponding to the inverse depth value of a pixel on an ...
1
vote
2answers
22 views

Probability more than 25% greater?

The random variable X is distributed N(60,64). The random variable Y is distributed N(52,36). Find the probability that a random observation from X is more than 25% greater than a random observation ...
1
vote
2answers
30 views

Estimate mean and variance for a truncated sample set

Assume there is a normally distributed random variable $X \tilde{} N(\mu, \sigma)$ I want to estimate $\mu$ and $\sigma$. So far the standard setting. Assume I am given a sample $(X_i)_{i=1}^N$ of ...
0
votes
1answer
30 views

What is the correct equation for “Normal distribution function of continuous random variable”?

I was reading a book and came across with a equation which gives the normal distribution function of continuous random variable. It was used in a software called ...
1
vote
1answer
11 views

T distribution problem

I will be using $t$-distribution to solve this problem. Specifically,the pooled variance test because both samples have size less than $30$,and both populations seem to have the same population ...
1
vote
0answers
50 views

Integrating a prob distr over the set of possible circles within an annulus

Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance ...
0
votes
2answers
66 views

Covariance and normal distribution

Let $X,Y ∼ N(0,1)$ be i.i.d. and let $U,V$ given by $U=aX+bY+c$ and $V=dX+eY+f$ have a bivariate normal distribution (here $a, b, c, d, e, f ∈ R$ with $ae − bd$ not equals to 0). (a) What is $Cov(X, ...
1
vote
2answers
26 views

What's $r$ going to be when you get the summation of $36$ Geometric $X_i$'s

Let $X_1,X_2,\ldots,X_{36}$ be a random sample of size $n=36$ from the geometric distribution with the p.d.f: $$f(x) = \left(\frac{1}{4}\right)^{x-1} \left(\frac{3}{4}\right), x = 0,1,2,\ldots$$ Now ...
0
votes
1answer
14 views

Definite integral of two dimensional normal distribution

I want to calculate the probability mass in a rectangular area for a two dimensional normal distribution $G(x;\mu,\Sigma)$ . Is it possible to do without numerical integration?
0
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0answers
14 views

What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A company has a one-year group life policy that divides its employees into two classes as follows: Class, Probability of Death, Benefit, Number in Class, A, 0.01, ...
0
votes
1answer
29 views

Derive a hypothesis test Z

Suppose a random sample of size 10 is taken from the random variable X which has the normal distribution with unknown mean $\mu$ and variance 4. You are requested to test the hypothesis $H_0:\mu = 0$ ...
0
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0answers
15 views

distribution of infinite sum of independent but non-identical normal variables

For $i=1,2,\ldots,n$, suppose $X_i \sim N(0,\Omega_{i})$, where $\Omega_{i}$ is of dimension $k\times k$. It is known that $\frac{1}{\sqrt{n}} \sum_{i=1}^{n} X_i \sim N(0, \overline{\Omega})$, where ...
0
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0answers
42 views

partical correlation in mixed case binomial and gaussian

For Gaussian mutlivariate distributions it is known, that zero partial correlation corresponds to conditional independence. Is there a same result if one of the variables has a binomial distribution? ...
3
votes
0answers
99 views
1
vote
2answers
28 views

Normal distribution with sample

I'm trying to figure out the best approach to this problem. I would assume that I can use the Central Limit theorem first and then a binomial cdf: Chocolate is packaged into jars using a computerized ...
1
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0answers
29 views

Gaussian Bayes Classification with dependent variables..

Gaussian Bayes Classification: two classes: $y \in \{-1,+1\}$ Dependencies for a vector of features ($x_1,x_2,x_3)$: $x_1=z,x_2=2z,x_3=t+3$, where $$P(z\mid y=+1) = \aleph(z;\mu_+,1),\qquad P(z\mid ...
0
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0answers
10 views

$\exists \mathcal{A},\mathcal{B}:X\sim \mathcal{A}\Rightarrow \frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$?

Does there exist a parametric distribution $\mathcal{A}$, such that: $X\sim \mathcal{A}\Rightarrow\frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$ for some parametric distribution $\mathcal{B}$ Where ...
3
votes
0answers
38 views

PDF of $X = \max\{X_1,X_2\}$, being $X_1$ and $X_2$ independent Normal distributed random variables

Let $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, what's the CDF of $X = \max\{X_1,X_2\}$? Both variables are assumed to be independent. I tried the ...
2
votes
2answers
52 views

If X is log-normal, is: $\frac{a}{\sqrt{b+cX}}$?

I am working for the first time with log-normal distributions and I want to verify whether the following statement is true. I am not sure whether all the properties of the log-normal distribution hold ...
1
vote
1answer
42 views

Probability - Normal Distribution, Heights of Women versus Men

The heights of young women aged $20$ to $29$ follow approximately the $\mathcal{N}(64, 2.7)$ distribution. Young men the same age have heights distributed as $\mathcal{N}(69.3, 2.8)$. Height is ...
0
votes
1answer
31 views

Minimum matching convolution (part II)

We assume we are working in $\mathcal{H}(\mathbb{R}^n)$, the space of real symmetric matrices. We define the partial order $\ge$ defined as $\Sigma_1\ge \Sigma_2$ iff $\Sigma_1-\Sigma_2$ is in ...
0
votes
1answer
19 views

Find $P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$ with a sample of $n=16$ and $X \sim N(50,100)$

If $X_1,X_2, ..., X_{16}$ is a random sample of size $n=16$ from the Normal Distribution $N(50,100)$, determine: $$P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$$ Okay well I know that ...
0
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0answers
10 views

how can we generate random numbers using skew normal distribution

I want to generate random numbers with skew normal distribution using rsn(). I can find the answer from the following link. how can we generate random numbers using skew normal distribution in ...
2
votes
1answer
25 views

Projection of Gaussian distribution along a vector.

Can anyone help me understand how to compute the projection of a 2D gaussian distribution along a vector. I intuitively realize that the projection will result in a 1D Gaussian, but I want to be sure. ...
0
votes
1answer
14 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, ...
0
votes
0answers
14 views

Multivariate gaussian and average covariance matrix

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...
0
votes
0answers
13 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
2
votes
1answer
31 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
0
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0answers
28 views

Quadrant probability of non-centric bivariate normal distribution

Suppose $(X,Y)$ has a bivariate normal distribuion with non-zero mean vector $\mu$ and covariance matrix $\Sigma$. What should $\mathbb{P}(X>0,Y>0)$ be? My attempt gives me an definite ...
0
votes
1answer
17 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
0
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0answers
28 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
-1
votes
1answer
32 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
6
votes
1answer
82 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
0
votes
1answer
18 views

If X + Y is truncated normal and X and Y are identitically (but not independently) distributed? What is the distribution of X and Y?

Let $(aX + bY)$ be a truncated normal and assume $X,Y$ are both identically distributed (but necessarily NOT independent) what is the distribution of $X$ and $Y$? More importantly can the pdf of $X$ ...
0
votes
1answer
34 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
1
vote
0answers
26 views

Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...