Questions on the Gaussian, or normal probability distribution, and related topics.
1
vote
1answer
19 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
36 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
27 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
16 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
36 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 ...
1
vote
0answers
20 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 ...
0
votes
0answers
26 views
Difference of normal r.v's. Please check my answer.
The length of a box is normally distributed with: $X \sim N(50;1.2)$
The length of a drill is normally distributed with: $W \sim N(49;1.2)$
Find the probability that a randomly selected drill will ...
0
votes
0answers
29 views
Convolution of logistic function and gaussian distribution
I am trying to solve the folowing problem:
$$\int \exp\left(-\frac{(x-u)^2}{2\sigma^2}\right) \log(1+\exp(ax + b)) \,dx$$
which I think is very complicated and there is no closed form solution(?)
...
2
votes
2answers
62 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
22 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
15 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
26 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
0answers
50 views
The expect value of normal distribution given that y is positive
Y is normal distributed with mean -2 and variance 25.Find the expect value of normal distribution given that y is positive.
I thought it in many ways, but still can not figure out.
0
votes
1answer
18 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
25 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 ...
5
votes
1answer
108 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 ...
0
votes
0answers
17 views
how to obtain the moments of skew-normal distribution?
the moment generating function of a skew normal distribution of random variable, z is defined as,
$$
M(t) = 2(e^{(t^2/2)})\Phi({{\delta}t)}
$$
where,
$\Phi$ refers to cumulative distribution function ...
1
vote
2answers
68 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
43 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
votes
1answer
20 views
0
votes
0answers
22 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
50 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
62 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
83 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:
$$\phi(x,y) = ...
0
votes
0answers
38 views
Probability that values from different normal distributions will be in a certain order?
I am looking at several normal distributions that describe the same metric from different sources (independent). I want to find the probability that they are in a certain order.
For example, I have ...
1
vote
1answer
35 views
Simple question on random variables and statistics
Let X1 and X2 be 2 random variables. X1 = 20. X2 = 30. Each of those has a standard deviation of 5.
If the random variables were normally distributed, what is the probability of getting such a ...
4
votes
2answers
105 views
Minimizing the expectation over a set, wrt to the Gaussian measure
I have recently read a proof [1] where, at the last step, the authors use an inequality which basically amounts to a lower bound on
$\int_\mathbb{R} \mathbf{1}_A(x)|x| \phi(x)dx$, where $\phi$ is the ...
0
votes
2answers
46 views
Calculating the MSE for assessment
Let $X_1, \ldots, X_n \sim \mathcal{N}(\mu, \sigma^2)$ be the sample, when $\mu$, $\sigma$ are unknown.
We suggest assessment for $\sigma^2$:
$$S^2 = \frac{\displaystyle\sum_{i=1}^n (X_i - ...
0
votes
0answers
30 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
30 views
Integral of the Normal Characteristic Function
The characteristic function of the $N$-variate Normal distribution is
$$\forall \mathbf{t} \in \mathbb{R}^N \quad
\psi(\mathbf{t}) \equiv \mathbb{E}\left(
e^{i\mathbf{t}X}\right) =
\exp \left( i{ ...
0
votes
0answers
41 views
How to normalize a set of vectors
I have a set of vectors $\displaystyle a_1, a_2,...,a_n$ and each of which has a dimension of $k$. How can I normalize the elements of these vectors to make them lie within $[0,1]$?
I was thinking ...
0
votes
1answer
23 views
Normal Distribution Calculating Probability
I am struggling with the following question:
A company which produces $1L$ beverages adjusts their machines in a way that
the filling quantity is normally distributed. The mean is ...
1
vote
2answers
29 views
Bound for erf function
For small $\epsilon \geq 0$ Is
$erf(\epsilon) \leq \epsilon$
Can somebody give me the hint
0
votes
1answer
42 views
Bound for the integral
Is there any way to bound the following integral
$$\int_{-(\epsilon+1)/\sigma}^{(\epsilon-1)/\sigma} \mathrm e^{-t^2/2}\,dt$$
1
vote
1answer
61 views
Derive the PDF of the log-normal distribution?
If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods:
(i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
5
votes
0answers
64 views
What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?
If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed.
What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
0
votes
0answers
23 views
Can one sample uniformly from the surface of an $n$-sphere of non-unit radius using normal r.v.'s?
One can sample coordinates of the surface of a unit radius $n$-dimensional sphere uniformly using the following method: independently generate a vector of $n$ standard normal random variables ...
0
votes
2answers
86 views
Indefinite integral of product of CDF and PDF of standard normal distribution
Is there a solution to:
$\int ^\infty _x \Phi(z) \phi(z) dz$
where $z$ ~ $N(0,1)$ and $\Phi$ and $\phi$ refer to the CDF and PDF?
Many thanks.
1
vote
1answer
21 views
Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution
I am trying to derive Chi-square distribution. The random variale is
$$ U^2=\sum_{i=1}^k X_i^2 $$
where $X$ is a random variable with normal standard distribution.
What is the distribution of ...
1
vote
3answers
50 views
Standardizing A Random Variable That is Normally Distributed
To standardize a random variable that is normally distributed, it makes absolute sense to subtract the expected value $\mu$ , from each value that the random variable can assume--it shifts all of the ...
0
votes
0answers
36 views
Can Bhattacharyya distance be greater than one?
I have two vectors, say $P$ and $Q$. I want to find the statistical overlap between two given that $P$ is my reference which I have modeled after Normal distribution and I have parameters for it. $Q$ ...
0
votes
0answers
23 views
Finding the joint distribution of 2 ratio of Gaussian random variables
Given independent normal random variables $X$, $Y$, and $Z$, I have the following ratios defined
$$
\begin{align}
r_1 &= \frac{x}{z}
\\
r_2 &= \frac{y}{z}
\end{align}
$$
The marginal ...
0
votes
0answers
18 views
Is $f_{\Theta|Z}(\theta|z)$ Gaussian when $Z = \theta^3 + V$, and given that $\Theta$ and $V$ are Gaussian?
$\Theta$ and $V$ are zero mean Gaussian random variables with variances $\sigma_\Theta^2$ and $\sigma_V^2$.
A third random variable $Z$ is defined as:
$$
Z = \Theta^3 + V
$$
Is ...
-1
votes
1answer
17 views
Bound for normal distribution
suppose $X$ is a standard normal distribution then what is the bound for
$Pr \{|X|\leq \epsilon \} $, where $\epsilon \geq 0.$
0
votes
1answer
140 views
special matrix in terms of its covariance matrix
How can we find a matrix $S\in \mathcal{M}_{n,n}$ and $Z\in \mathcal{M}_{n,m}$ whose $n$ entries of the $i^{th}$ column $Z_i$ are correlated $Z_i \sim \mathcal{N}(0,S)$ where $S \in \mathcal{M}_{n,n}$ ...
2
votes
1answer
114 views
Variance of $\exp(-x)$
Hi I have been struggling to find the variance of the $\exp(-x)$ in terms of $\exp$.
For the function Y = exp (-x) where X is N (0,1) show that the variance of Y = $\exp(\exp-1)$
This is what I ...
3
votes
3answers
85 views
$E(1/(1+x^2)) $under a normal distribution
I want to know as mentioned in topic $E(1/(1+x^2))$ under a normal distribution $N(0,1)$. If it's not analytical, are there any bounds that are possible?
So basically,
...
0
votes
1answer
35 views
Normal distribution bound
Let $X$ be a random variable which follows normal distribution.
Is True that
$Pr[|X|\leq \epsilon] \leq \epsilon$ for all $\epsilon \geq 0$.
0
votes
1answer
30 views
Normal distribution in equality
Let $p(x)=a_1x_1+a_2x_2+. . . .a_n x_n$ be a polynomial such that
$\sum_ia_i^2=1,$ each $x_i \sim N(0,1)$. then we know that $p(x) \sim N(0,1)$.
How can we bound $\Pr_{x\in ...
0
votes
2answers
29 views
probability normal distribution
A model for the movement of a stock supposes that if the present price of
the stock is s, then after one time period it will be either (1.012)s with
probability 0.52, or (0.99)s with probability ...
