Questions tagged [normal-distribution]
This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.
7,403
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Compute posterior distribution given Gaussian likelihood and Normal-Gamma prior
Gaussian distribution likelihood
$$p(y|μ,τ) = N(y; μ,τ^{−1}) =\frac{\sqrt{τ}}{\sqrt{2π}}e^{-\frac{(y−μ)^2τ}{2}}$$
where μ,τ are unknown.
Normal-Gamma distribution as the conjugate prior
$$
NormalGamma(...
- 214
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32
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Probability Question - Random Variables
Suppose I have 300 daily returns that come from one of two distributions. The probability it comes from distribution 1 is 50% and the probability it comes from distribution 2 is 50% (i.e. equally ...
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1
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Applying Central Limit Theorem to an exponential distribution - how big should sample size be?
For an exponential distribution, in order for the sampling distribution of its mean to be well approximated by normal distribution (via central limit theorem), how big should a "typical" ...
- 1,540
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40
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Find the probability of intersection of time intervals
There is an object that is always in one of two states - $A$ or $B$. The duration of its stay in both states is given by a normal distribution whose parameters are $σ_1, μ_1$ for state $A$, and $σ_2, ...
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25
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For what values of $\mu$ are the Poisson and Gaussian distributions approximately equal?
Let $\displaystyle P_P(n,\mu)=\frac{\mu^n}{n!}\,e^{-\mu}$ and $\displaystyle P_G(x,\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}\,\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\quad$ where $\sigma=\sqrt{\mu}$...
- 238
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43
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What distribution am I sampling from?
Assume that $X_1$ and $X_2$ are random variables with Gaussian joint distribution
$$ \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim \mathcal{N}(
\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix},
\begin{...
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1
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27
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Diffrence in probability distributions of sepertaed groups
If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and ...
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In the 1992 presidential election Alaska's 40 election districts averaged 1886 votes per district for President Clinton The standard deviation was 600 [closed]
In the 1992 presidential election, Alaska's 40 election districts averaged 1886 votes per district for President Clinton. The standard deviation was 600. (There are only 40 election districts in ...
0
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1
answer
31
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Proof of Independence of Sample mean and sample variance
This concerns Section 7.8.2 in the book A First Course in Probability by Sheldon Ross 10th Edition.
Section 7.8.2 says that :
Let $X_1, X_2.\ldots X_n$, be independent normal random variables each ...
- 283
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9
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Does "independent variables X and dependent variable Y are jointly gaussian" means "the residual term has 0 conditional mean"?
I saw the following statement in my lecture note:
"The data generation process is $y = x+\epsilon$, whereas in the regression we run y on x so the regression model is $y = \beta_{OLS} x+e$, the ...
- 87
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1
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28
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Compute the probability of multivariate Gaussian distribution
I am reading a proof from a paper:
Zhu, Sicheng, Xiao Zhang, and David Evans. "Learning adversarially robust representations via worst-case mutual information maximization." International ...
- 127
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1
answer
17
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Sampling Normal Distribution; Box-Muller, Inverse Transform, Rejection, Approximations?
I assume $X\sim\mathcal{N}(\mu,\sigma)$ and wish to sample values but I am confused about different approaches and concepts that seem to be relevant for this problem.
It appears to me that this ...
-2
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1
answer
35
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Exponential distribution from Standard normal distribution
How do we get part (c)? I tried looking for relationships between the chi square distribution and the exponential distribution but couldn't find anything.
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Conditional density calculation
If we know the density of a vector $x$ is $f(x)$, which could be multivariate standard normal $N(0,I_r)$, and we know the conditional density of $f(y|x)$, which is multivariate normal $N(Ax,I_T)$ with ...
- 393
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23
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Posterior Distribution and James Stein Estimator
Assume $\mathbf{\mu }\sim N\left( 0,I_{r}\right) ,$ where $I_{r}$ is a $%
r\times r$ identity matrix, and $\mathbf{y}|\mathbf{\mu }\sim N\left(
\mathbf{A\mu },I_{T}\right) ,$ where $\mathbf{A}$ is a $...
- 393
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votes
1
answer
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Calculate $\mathbb{E}[Z\Phi(aZ+b)^n]$
I am considering the integral $\mathbb{E}[Z\Phi(aZ+b)^n]$, where $Z\sim\mathcal{N}(0,1)$, $\Phi$ is the cdf of a standard normal distribution, $n\in \mathbb{N}$, $a,b\in \mathbb{R}$. In the following, ...
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The product of two random variables, one is Gaussian distribution, the other one is uniform sampling from $SO(3)$ group [closed]
Suppose we have two random variables, $X,M$, $X\sim\mathcal{N}(\mu,\Omega)$, and $M\sim Uniform(SO(3))$, is the product $MX$ a Gaussian or some other named random variables?
Thanks!
- 525
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25
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Distributions of a product and of sums of products of iid, standard normal random variables.
It is known how to compute the distribution of a product of independent random variables.
The results boils down to evaluating the inverse Mellin transform of a product of Mellin transforms of the ...
- 10.7k
4
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1
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129
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Calculate $E[\Phi(aX+b)^n]$
I am working on a project, where this expected value shows up again and again, where $X \sim N(0,1)$ and $\Phi$ is the cdf of a standard normal distribution.
$$E[\Phi(aX+b)^n]$$
However, I have not ...
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1
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1
answer
169
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Implied Correlation from Gaussian Copula
I am building a spreadsheet model that allows marginal distributions to be correlated together using a Gaussian copula (with prescribed correlation matrix).
The inputs into the model are the ...
- 739
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answers
25
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Decoupling $n$ Gaussian random variables using linear algebra
For a bivariate normal distribution $\begin{pmatrix} X \\ Y \end{pmatrix}$ with mean $\begin{pmatrix} \mu_X \\ \mu_Y\end{pmatrix}$ and variance $\begin{pmatrix} \sigma_X^2 & \rho \sigma_X \sigma_Y ...
- 400
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14
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A bit of confusion with the Gaussian measure and pushforward by a random vector
Let $d\mu$ be a centered Gaussian measure on $\mathbb{R}^n$ with the covariance matrix $\sigma$ and $X : \mathbb{R}^n \to \mathbb{R}^n$ be any measurable mapping. Or we can simply regard $X$ as a ...
- 6,808
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16
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Stationary distribution of Markov chain with continuous state space
I'm considering a Markov process with a continuous state space. Let $V(x)$ be a differentiable function, $\Delta t$ a fixed time step, and, at every step, set
$$x_{n+1}= x_n-\alpha \frac{dV}{dx}\Big|_{...
- 1,025
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1
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21
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$\Sigma$ norm in Normal Distribution
My professor wrote down this in class:
Let $D=N(\mu,\Sigma)$ be a normal distribution, then $$\log(p_D(d))=-\frac{1}{2}||d-\mu||^2_\Sigma+\text{const.}$$
Here I don't completely understand from where ...
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31
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Gradient of a Gaussian Quadratic Form
For research purposes, I am solving a particular case of an optimal control problem. I need to compute
$$\nabla_{u_t} (x_t^TC^TQ_tv_t) $$
where $x_t \in \mathbb{R}^n$, $ C \in \mathbb{R}^{q \times n}...
1
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1
answer
47
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Is $X Y/|Y|$, where $X, Y \text{ i.i.d. } \sim N(0, 1)$, normally distributed?
In this answer, it was shown that the product of two i.i.d. standard normal r.v.s is not normally distributed. What if we scale one of them? In other words, if $X, Y \text{ i.i.d } \sim N(0, 1)$, what ...
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1
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Please help me derive the formula for upper bound for one sided confidence interval $\bar{x} + z_{\alpha}(\frac{\sigma}{\sqrt{n}})$?
I want to derive for myself the known formula for the upper bound for one sided confidence interval $\bar{x} + z_{\alpha}(\frac{\sigma}{\sqrt{n}})$ for mean $\mu$ for a sample of size $n$ from a ...
- 1,540
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24
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determine if the variance estimator is consistent and efficient of a normal distribution
I have the stimator $σ^2_{n}=1/n⋅(∑_ {i=1}^{n}(Xi−µ_{0})^2$ from a normal distribution with mean μ (known) and variance $σ^2_{n}$ (unknown) I have to determine if the stimator is:
Efficent (Cramer-...
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22
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Quadratic form plus linear for is a quadratic form
I am searching for a way to reduce the following expression:
$$
\frac{1}{2}(y-\mu)^T\Sigma^{-1}(y-\mu) - x^TP^Ty= \frac{1}{2}(y-\tilde{\mu})^T\tilde{\Sigma}^{-1}(y-\tilde{\mu})
$$
I need to find a ...
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16
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Get $b$ value that minimized error
Let's say I have a list of values that I got sampling the normally distributed random variables $[X_i]_{i=1}^{i=N}$ once (as in 1 value from each RV), and I know that $X_i\sim N(b\cdot k_i,\sigma^2)$ ...
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1
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48
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Does a bounded section of the normal distribution converge to the uniform distribution?
It was once asked on CrossValidated whether the normal distribution converges to a uniform distribution when the standard deviation grows to infinity. (The answer was no.) I am curious about a related ...
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1
answer
33
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Covariance matrix of an anisotropic normalized Gaussian random vector
Let $\xi \sim \mathcal{N}(0, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^n$. I would like to calculate the covariance matrix of the normalized vector $\frac{\xi}{\lVert \xi \rVert}$, i.e.:
$$ ...
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0
answers
36
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Fourier Transform of $\exp\left[-e^{-k^2}\right]$
I want to calculate the inverse Fourier transform of $\exp\left[-e^{-k^2}\right]$. One of the ways I can imagine is to expand the exponential in a series
$$
\exp\left[-e^{-k^2}\right] = \sum_{n=0}^\...
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0
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35
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What is the minimum score to get a scholarship?
The test scores of an entrance examination are normally distributed with a mean $230$ and standard deviation $80$, the cutoff for admission was $310$ what % of examinees got admission, further, the ...
1
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0
answers
37
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Show that $(M_1,\cdots,M_p)$ and $(N_1,\cdots,N_q)$ are independent
Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space and $X_1,\cdots,X_n:\Omega\to\mathbb{R}$ be independent normally distributed random variables. Show that if $M_1,\cdots,M_p,N_1,\cdots,N_q\in\...
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22
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Closed form expression for the minimal width interval that satisfies coverage on log-normal distribution
I am interested in finding the minimum width interval $(b_l, b_u)$ for $b_l,b_u \in \mathbb{R}$ that obtains a coverage level of $\alpha \in (0,1)$ for the log-normal distribution. The CDF of the log-...
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21
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Expectation of the pseudoinverse of a complex Gaussian matrix with non identically distributed columns
Let us define the $M \times N$ matrix $\boldsymbol{C}=\left[\boldsymbol{c}_1 \cdots \boldsymbol{c}_N\right]$, where $\boldsymbol{c}_n \sim \mathcal{CN}\left(\boldsymbol{0}, \boldsymbol{R}_n\right)$ (i....
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20
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+50
Confusion with Complex Gaussian process with Auto-covariance
I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ...
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4
votes
2
answers
111
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fully customizable periodic function?
I am looking for a bell-shaped periodic function f(x) with parameters a and b, with following characteristics:
( not sure if such function already exists or one can formulate one ) :
oscillating ...
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0
answers
36
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How to prove $x^T\Sigma x$ is quadratic?
How to prove $x^T\Sigma x$ is quadratic? where $\Sigma$ is covariant varible(symmetric matrix)
I did simple calculation with 3x2 matrix , 2x2 covariance, 2x3 matrix. the result is 3x3 matrix. But how ...
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Mean of sample generated by Exp and Norm
I have the following scenario: i have a next event simulation in which I generate an arrival process in time slot [8:00 am, 11:00 am]. The inter-arrival time is assumed exponential with arrival rate 2 ...
- 99
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23
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Does the x in the z score need to be distributed as a Gaussian? [closed]
I was wondering if the $x$ in the expression $$z=\frac{{x-\mu}}{\sigma}$$
must be distributed as a gaussian or not. None of my books specify that is required so I was wondering if the z score test ...
- 3
3
votes
2
answers
121
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A tight upper bound for Gaussian integral.
Consider two positive real number $\mu$ and $\sigma$. Let $m = 1, 2, \ldots$ be the natural number, I want to find a tight upper bound for the following part Gaussian integral:
$$\int_0^{\infty} \exp \...
- 343
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votes
1
answer
58
views
Expected value conditioned on highest signal
Let $v\sim N(0,\sigma_v^2)$, let $z, y_1, \dots, y_n\sim N(v, \sigma^2)$ be i.i.d stochastic variables.
Calculate $E[v\mid z, z \geq y_i \ \forall i \in \{1,\dots , n\}]$
Said in another way, then ...
- 77
1
vote
1
answer
222
views
The correct way to integrate the variational free-energy formula
Lets define the following probability density distributions as:
$$
\begin{align}
p(θ) &= N(θ; 0,1) \\
q(θ) &= N(θ; μ,σ^2)\\
p(y|θ,x) &= N(y; θx,σ_n^2)
\end{align}
$$
where $N(x; m,v)$ ...
- 214
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0
answers
12
views
Finding mutual information in discrete linear partial observation stochastic process
I have one basic question maybe is not to hard for you but I am a bit confused. Let our system be like this:
\begin{align}
X_{k+1} &= A_k X_k + W_k \\
Y_k &= C_k X_k + V_k
\end{align}
where $...
- 1
0
votes
0
answers
21
views
Overall minimizer of the minus log-likelihood of joint Gaussian distribution
Consider the following model. There is a DAG $D_0$ whose p nodes correspond to random variables $X_1,...,X_p$: assume that
$$X_1,...,X_p \sim N_p (0, \Sigma_0) \text{ with density } f_{ \Sigma_0} (\...
- 511
1
vote
0
answers
14
views
Norm inequality for a linear combination of Gaussian vectors
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 65
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votes
0
answers
13
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Positivity of posterior variance
Given a gaussian process with a kernel k, we know the posterior variance after conditioning on data $D_n = \{x_1, \cdots, x_n\}$ is given by
$$
k(x,x) - k_n^T(x) K_{nn}^{-1}k_n(x),
$$
where $k_n(x) = (...
- 164
1
vote
0
answers
46
views
Joint Gaussians of three random variables
Suppose that
$(X,Y)$ are joint Gaussian with $Cov(X,Y)=\theta_1$,
$(Y,Z)$ are joint Gaussian with $Cov(Y,Z)=\theta_2$.
Are $(X,Z)$ also joint Gaussian? What is $Cov(X,Z)$?
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