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

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5
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3answers
271 views

Central Limit Theorem Definition

My friend and I have a bet going about the definition of the Central Limit Theorem. If we define an example as a number drawn at random from some probability density function where the function has a ...
5
votes
1answer
212 views

A case of the central limit theorem

I want to show that $$\frac{\sum_{k=1}^N X_k}{\sqrt{\sum_{k=1}^N X_k^2}} \overset{N\to\infty}{\to} \mathcal{N}(0,1)\text{ in distribution,}$$ where $X_1,X_2,\ldots$ is a sequence of iid random ...
4
votes
0answers
126 views

Characterization of the law of a stochastic process by its finite dimensional distributions

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space. Let $(X_t)_{t \in [0,T]}$, $(Y_t)_{t \in [0,T]}$ (real-valued) centered Gaussian processes such that the finite dimensional distributions ...
3
votes
2answers
179 views

$X,Y$ i.i.d., $X$ and $(X+Y)/\sqrt{2}$ have the same dist., then show that $X$ has a normal distribution

I am trying to show that $X$ is a standard normal (in distribution) by applying the Lindberg's version of the central limit theorem to a sequence always equal to $X$. In order to do that, I need to ...
1
vote
1answer
227 views

Projections of multivariate normal distribution

Given a random vector X with the multivariate normal distribution F(X), we know that, for two vectors a and b, the projections $A=\sum_j a_j X_j $ and $B=\sum_i b_i X_i $ are univariate normal. I'm ...
0
votes
1answer
73 views

Regression vs. Normal Distribution

I have to estimate something using historical data. Should I find the equation of the curve of best fit to estimate? Or use a confidence interval, standard deviation, and a z-score to calculate it? ...
2
votes
2answers
149 views

normal distribution, two independent random variable

if $X$ and $Y$ are independent normal distribuited random variables and $T=2X-Y-1$ and $E[X]=E[Y]=1$ and $Var(X)=Var(Y)=4$, what is $Var(T)$? I get $E[T]=E[2X-Y-1]=2-1-1=0$, but i don't know how to ...
2
votes
0answers
31 views

The distribution of the result of Monte-Carlo method

For example, if I want to determine the probability of getting tails when tossing a coin. By Monte-Carlo method, I toss the coin 1000 times and got 600 tails. As I know the distribution of the result ...
1
vote
0answers
72 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
1
vote
1answer
459 views

Relation chi-square and student-t distribution

First I want to prove that the sum $Y_1+...+Y_n$ where $Y_i=X_i^2$ and $X$ is standard normally distributed has density $f_n(x)=c_n x^{n/2-1}e^{-x/2}1_{x>0}$ I do not want to derive it, I would ...
1
vote
1answer
48 views

distribution of Brownian Motion involving integral

What is the distribution of $\int_{t}^{T} W(s)ds$? Given that W(t) is brownian motion. So far, I have the following, $\int_{t}^{T} W(s)ds$ = $(T-t)W(t) + \int_{t}^{T} (T-s)dW(s)$ Also, ...
1
vote
1answer
48 views

Normal distribution Q

Human heights are one of the many biological random variables that can be modelled by the normal distribution. The average height of Canadian women aged 18 and older is 163cm, while the average height ...
1
vote
1answer
28 views

Changing bounds of integrals

If I have: $$\int_{L}^{\infty }e^{\dfrac{-(x-\sigma \sqrt{T})^2}{2}}\,dx$$ let $y = x - \sigma \sqrt{T}$ $$\int_{L - \sigma \sqrt{T}}^{\infty }e^{\dfrac{-y^2}{2}} \, dy$$ Why does the lower bound ...
4
votes
1answer
84 views

Is there a real number, that is proven to be normal in every base, whose digits can be enumerated by an algorithm?

To clarify, I mean every natural number base $b$ where $b \geq 2$. If so, what is the algorithm to generate the number (and what is the number, if it has a name)?
1
vote
1answer
541 views

probability of sample variance lying between given values

Let $X_1,\ldots,X_n$ be a random sample of size $n = 10$ from a population which is Normally distributed with mean $= 48$ and variance $= 36$. What is the probability that the sample variance of such ...
2
votes
1answer
87 views

Square of a normal distribution

Let $Z$ have a normal distribution with mean $\mu$ and variance $1$. Show that $Z^2$ is a continuous random variable and find its p.d.f. I really don't know what to do with this... I tried working ...
1
vote
1answer
64 views

Importance of estimating $\sigma^2$ in linear Statistical model

Statistical model for Complete Randomized design $y_{ij} = \mu + \tau_i + \epsilon_{ij}$ where, $i$ denotes treatment and $j$ denotes observation. $i=1,2,...,k\quad and \quad j=1,2,..., n_i$ ...
0
votes
1answer
124 views

Conditional probability of the sum of r.v.

I have $n$ independent random variables $X_i$ with known PDF and CDF (say, Normal, but not necessarily with the same parameters). Given $U_1, U_2 \subseteq \{1,...,n\} $ such that $U_1 \cup U_2 = ...
1
vote
2answers
94 views

How do you get hyperbola by squaring a bell?

This is a Gaussian bell (aka normal distribution). Its square, I belive looks the same. Yet, I see that chi-square distribution, which is a sum of k such bell squares, looks like Take a look at ...
1
vote
1answer
289 views

Computing the expected value of a matrix?

This question is about finding a covariance matrix and I wasn't sure about the final step. Given a standard $d$-dimensional normal RVec $X=(X_1,\ldots,X_d)$ has i.i.d components $X_j\sim N(0,1)$. ...
1
vote
1answer
107 views

Integral involving normal densities

I am trying to solve the integral $$I(y)=\int_{\mathbb R}f(x,y)g(x)dx,$$ where $f(x,y)$ is the bivariate normal density with known mean $(\mu_1,\mu_2)$ and covariance matrix $\Sigma$ , and $g(x)$ is ...
0
votes
0answers
82 views

Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin

I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
0
votes
1answer
28 views

Splitting multivariate normal into individual (correlated) components

I have a multivariate normal variable $X$ with mean zero and variance $\Sigma$. I would like to write every component $X_i$ of $X$ as: $$ X_i = \phi_i Z_i $$ where $\phi_i$ is a scalar and $Z_i$ is a ...
0
votes
1answer
271 views

How do I prove Poisson appraches Normal distribution

I want to prove why the mean and variance of a $\operatorname{Poisson}(\lambda)$, is different when the time index approaches infinite (it's approximated by the mean and variance of a Normal). For ...
0
votes
0answers
160 views

Central Limit Theorem for Dependent Non-Identical Random Variables.

If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$. How do we apply ...
2
votes
2answers
306 views

Distributing $m$ balls into $n$ urns with no urn left empty. [duplicate]

If $m \geq n$, how many different ways are there of distributing $m$ indistinguishable balls into $n$ distinguishable urns with no urn left empty? I have no idea how to even start with this so any ...
0
votes
0answers
93 views

Multivariate Normal Product Distribution

I am looking for multivariate case of a distribution of a product of two normally distributed variables X and Y. The variables are independent. Something similar to this: ...
2
votes
0answers
27 views

Multivariate Distribution Question?

If $(X,Y)$ have the following joint distribution: $$f_{X,Y}(x,y) = \begin{cases} 2 f_X(x)f_Y(y) & \text{if }xy>0 \\[6pt] 0 & \text{otherwise} \end{cases} $$ where $f_X(·)$ and $f_Y(·)$ ...
1
vote
0answers
79 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
0
votes
1answer
36 views

Text length probability function that peaks for a an average length

I am looking at web page to tell what is its content (the main text part) and its title. I can estimate if I am looking at a title or content by the page's semantics, but wanted to add a rules that ...
1
vote
1answer
242 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
6
votes
1answer
411 views

How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
1
vote
2answers
112 views

Standard Brownian Motion

Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$. Can anyone give me some hint to start the ...
5
votes
1answer
54 views

Why does adding 3 random decimals in the range [-1,1] give a normal dist with std. dev 1?

I've used Math.random()*2-1+Math.random()*2-1+Math.random()*2-1 many times in the past to get normally-distributed random numbers with a standard deviation of 1. ...
0
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1answer
29 views
1
vote
0answers
96 views

Probability that a point from one normal distribution is greater than points taken from several other distributions?

I am looking at several normal distributions that describe the same metric from different sources (independent). I want to find the probability that each is greater than all the others. For example: ...
1
vote
1answer
80 views

integral of normal distribution

how to do this integral: $$ \mathop{\int\int}_{y+2x>0} x y \frac1{2\pi\sigma_x\sigma_y}e^{ -\frac{(x-\mu_x)^2}{2\sigma_x^2}}\cdot e^{ -\frac{(y-\mu_y)^2}{2\sigma_y^2}} dx dy$$ Both x and y are ...
4
votes
2answers
100 views

Central Limit Theorem. How to apply to the task.

The research showed that the probabilities of 3, 4, 5, 6 and 7 cars broken on one day are 0.3, 0.4, 0.2, 0.08, 0.02 respectively. If 221 car broke in 50 days, does it show that more cars break than ...
0
votes
1answer
119 views

how to do this integral: $ \int_{0}^{\infty} \int_{0}^{\infty} x y \phi(x, y) dx dy$

how to do this integral: $$ \int_{0}^{\infty} \int_{0}^{\infty} x y \phi(x, y) dx dy$$ where $ \phi(x,y)$ is a general pdf of bivariate normal distribution, that is: $$\phi(x,y) = ...
0
votes
0answers
54 views

Probability that values from different normal distributions will be in a certain order?

I am looking at several normal distributions that describe the same metric from different sources (independent). I want to find the probability that they are in a certain order. For example, I have ...
1
vote
1answer
91 views

Simple question on random variables and statistics

Let X1 and X2 be 2 random variables. X1 = 20. X2 = 30. Each of those has a standard deviation of 5. If the random variables were normally distributed, what is the probability of getting such a ...
4
votes
2answers
191 views

Minimizing the expectation over a set, wrt to the Gaussian measure

I have recently read a proof [1] where, at the last step, the authors use an inequality which basically amounts to a lower bound on $\int_\mathbb{R} \mathbf{1}_A(x)|x| \phi(x)dx$, where $\phi$ is the ...
0
votes
2answers
72 views

Calculating the MSE for assessment

Let $X_1, \ldots, X_n \sim \mathcal{N}(\mu, \sigma^2)$ be the sample, when $\mu$, $\sigma$ are unknown. We suggest assessment for $\sigma^2$: $$S^2 = \frac{\displaystyle\sum_{i=1}^n (X_i - ...
0
votes
1answer
52 views

Homework Help. Probability Density Functions.

$X$ is $N(10,1)$. Find $f(x|(x-10)^2 < 4)$ This is a homework question. I can only figure out that X is normally distributed with mean 10 and variance 1. Can you please explain what is meant to ...
2
votes
0answers
67 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
0
votes
1answer
45 views

Normal Distribution Calculating Probability

I am struggling with the following question: A company which produces $1L$ beverages adjusts their machines in a way that the filling quantity is normally distributed. The mean is ...
1
vote
2answers
100 views

Bound for erf function

For small $\epsilon \geq 0$ Is $erf(\epsilon) \leq \epsilon$ Can somebody give me the hint
0
votes
1answer
46 views

Bound for the integral

Is there any way to bound the following integral $$\int_{-(\epsilon+1)/\sigma}^{(\epsilon-1)/\sigma} \mathrm e^{-t^2/2}\,dt$$
1
vote
1answer
1k views

Derive the PDF of the log-normal distribution?

If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods: (i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
6
votes
0answers
225 views

What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated ...