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.
7,779 questions
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Finding a CI for variance using Chi squared, when mean is known and samples are normal but not canonized
I've been meaning to find a $(1-\alpha)$ CI for the variance of a normal distribution given $n$ samples $X_1,X_2,\dots, X_n$, where the mean is known. Where in class we've been introduced to a ...
- 497
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23
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Total variation distance of discrete gaussian and discrete uniform($-\lceil \sigma\rceil+1, \lceil \sigma\rceil$)
I want to find the statistical distance (Total variation distance) between a discrete gaussian centered in $0$ and standard deviation $\sigma$, and a discrete uniform with set $(-\lceil \sigma\rceil, \...
- 109
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24
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Prove that $Y=\frac{1}{\sigma^2}\sum\limits_{i=1}^n(X_i-\mu)^2$ and $\bar{X}$ are not independent
Given that $X\sim N(\mu,\sigma^2)$, $X_1，X_2,...,X_n$ is a set of samples and $\bar{X}=\frac{1}{n}\sum\limits_{i=1}^nX_i$, how to prove that $Y=\frac{1}{\sigma^2}\sum\limits_{i=1}^n(X_i-\mu)^2$ and $\...
- 21
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2
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115
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Sum of Cauchy distributions
Consider a sum of Cauchy distributions
$$
L(x,\eta)=\frac1{\pi}\frac{\eta}{x^2+\eta^2}.
$$
centred at $x=0$ and with widths $\eta_n=n$, for $n=1,2,\ldots$. Specifically
$$
G_N(x)=\sum_{n=1}^Na_nL(x,n),...
- 1,223
8
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2
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150
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Asymptotes of an integral
Given the function
$$f(x,y)=\int_0^{\infty } \frac{1}{\sqrt{u}} e^{-\frac{x^2}{u}} e^ {-\frac{(y-u)^2}{2}} \, du$$
for $x>0$, I am wondering if it is possible to find an explicit expression for it, ...
- 305
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2
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276
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Limit $\mathbb E[|(\text{Poi}(\mu)-\mu)/\sqrt{\mu}|^{k}]\, /\,\mathbb E[|\text{N}(0,1)|^{k}]$ as $\mu=k^{3}$ and $k\to\infty$
Consider a Poisson random variable $X \sim \text{Poisson}(\mu)$ and a standard normal random variable $Y \sim \text{N}(0,1)$. We're interested in comparing their normalized moments.
For any positive ...
- 4,995
3
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54
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Let $X_1, \dots, X_n$ be normal random variables. What is the distribution of $\sum X_i^2 \mid \sum X_i$?
Let $X_1, \dots, X_n$ be i.i.d. $N(\mu, \sigma^2)$ random variables. Suppose that I have observed the sum, $\sum_{i=1}^n X_i$, and I want to know the distribution of the sum of the squares, $\sum_{i=1}...
- 2,435
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18
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Is a least squares minimization equivalent to taking the product of the gaussians described by the data?
Suppose you have $N$ independent measurements of linear combinations of a multivariate normal vector $X$ such that measurement $D_i$ is $A^T_iX = \mu_i \pm \sigma_i$. One way to estimate the mean $\mu$...
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1
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41
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Generalised Laplace/Gauss transform of $\frac{\exp(-ax)}{x}$
Let $a>0$ be a constant and suppose $x\in\mathbb R_+$, how to prove：
$\frac{\exp(-ax)}{x}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^\infty \exp(-x^2t^2-\frac{a^2}{4t^2})dt$ ?
This seems like the ...
- 65
1
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0
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28
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Auto-correlation Calculation
I am solving a problem where the noise in the system is given by a Wiener process
$$\Phi_{L}(t) = \int_0^t \Phi'_{L}(\tau)d\tau $$
where $\Phi'_L(t)$ is modelled as a zero-mean white Gaussian process ...
- 41
2
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24
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Convergence to Independent Gaussian Processes
Suppose $(X_n(t):t \in [0,1])_{n \in \mathbb{N}}$ and $(Y_n(t):t \in [0,1]))_{n \in \mathbb{N}}$ are two sequences of stochastic processes taking values in $C([0,1])$ (equipped with uniform topology). ...
- 139
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Fixed-Length Confidence Interval Using Varying Sample Size [closed]
I had a question regarding an exercise aimed at creating bounded length confidence intervals using a varying sample size from which we compute the sample variance for a fixed number. More specifically,...
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Distribution of the Square of the Minimum of Two Standard Normal Random Variables // Proving $Z^2 \sim \chi_1^2$ for Z=min(X,Y) [closed]
Let $X$ and $Y$ be i.i.d. $N(0,1)$ random variables, and define $Z = \min(X, Y)$. Prove that $Z^2 \sim \chi_1^2$.
Let X and Y be independent and identically distributed standard normal random ...
- 1
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8
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How to relate a sequence following an error fuction and another sequence derived from the first sequence using their standard deviations?
I am an engineering student and want to derive a reasonable relation between a sequence derived from a normal distribution and the other seqence derived from the first sequence.
I have a sequence of ...
- 11
1
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0
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24
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Show a random walk bridge from $-N$ to $N$, with standard normal steps, converges in law to a Brownian bridge on $(-1,1)$.
Consider a random walk on $[-N,N] \cap \mathbb{Z}$, with Gaussian step disribution $N(0,1)$, conditioned on being $0$ on the boundary. Denote this random walk by $h_N$. I have an exercise which I can'...
- 421
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17
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Berry-Esseen Bounds for Summation of CDF
Consider the Binomial Random Variable $X \sim Bin(n, p)$.
I want to approximate $\sum_{x = 0}^a F(x)$ using the Normal, where $F(x)$ denotes the $Bin(n, p)$ CDF.
We know that using Berry-Esseen ...
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22
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Sample complexity of covariance matrix estimation of a Gaussian random variable (with explicit constants)
I'm looking for an explicit bound for the number of samples required to estimate the covariance matrix of a Gaussian distribution. In https://arxiv.org/pdf/1011.3027v7 (end of page 31), the following ...
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1
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21
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Ratio of Gaussian Quadratic Form
Let $A \in \mathbb{R}^{d\times d}$ be a symmetric and PSD matrix. Let $Z \sim \mathcal{N}(0,\mathbb{I}_d)$ be an istropic Gaussian random variable. I am interested in an upper bound on
\begin{equation}...
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25
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Standard deviation tells how a data is spread around mean. Then what is 1 SD (68%), 2SD (95%)?
I am learning statistics from scratch. I understand that SD would tell how datapoints are spread around mean. However, I do not understand how it is useful in finding outliers. Especially, these two ...
- 1
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21
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How Does Random Cropping and Resizing Affect the Distribution of Features in Image Classification
I am working on an image classification task using deep learning. My network extracts features from an image using a feature extractor and then passes these features to a classifier for image ...
- 1
2
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3
answers
104
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Distribution of a Uniform between Two Gaussians
I am having some problems with this question:
given two IID standard gaussians $X_1, X_2\sim\mathcal{N}(0, 1)$, let $Y_1, Y_2$ the correspondent order statistics such that $Y_1 < Y_2$, then let $Z\...
- 123
1
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1
answer
66
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Using the symmetry of the normal to find $E(Z|Y)$ given $Z\sim N(0,1)$ and $Y=Z^2$
I know that this question has been answered before; namely here.
I'm really struggling to understand the formulation of how this answer was reached. Specifically, as someone pointed out in the ...
- 141
1
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1
answer
45
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If $p(a | b)$ and $p(b)$ are Gaussian, does it imply that $p(b | a)$ or $p(a, b)$ are Gaussian too?
The title contains the whole question:
If we only know that $p(a | b)$ and $p(b)$ are Gaussian, does it imply that $p(b | a)$ or $p(a, b)$ must be Gaussian too?
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53
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Understanding mathematics of heuristic-based outlier detection: concerns about scoring, weighting, and validity
I am trying to understand the mathematics and methodology behind a newly published outlier detection algorithm in the Computer & Security journal. This algorithm uses heuristic-based approaches, ...
- 37
1
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2
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69
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Computing an expected value of a product of elements of the sample covariance matrix in two different ways with conflicting results.
Let $ N, T \ge 1 $ be integers and
let $C:= {\tilde C} \cdot {\tilde C}^T $ be a positive definite matrix of size $N \times N $:
Then let $ c:= X \cdot X^T $ where $X = {\tilde C} \cdot Z $ with
$$
...
- 11.8k
4
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1
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Why do powers of $\operatorname{sinc}$ converge to a Gaussian function?
I am looking to find an intuition for how powers of the $\text{sinc}$ function behave. Just for context, the $\text{sinc}$ function I'm looking at is the "unnormalized" one:
$$
\text{sinc}(x)...
- 1,043
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0
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45
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Mix of Gaussian laws and Cauchy laws
Let
$n \in \mathbb{N}$
$X_1 ... X_n$ are independent and follow a gaussian law $\mathcal{N}( m, \sigma^2)$
$Y_1 ... Y_n$ are independent and follow a gaussian law $\mathcal{N}( m, \sigma^2)$
$(X_i)_i$...
- 11
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39
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what is the function if I add a multiplier to a gaussian distribution
Gaussian distribution:
$$
y = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right)
$$
we know that it has only 2 parameters and it has a peak $y_{\text{max}} = \dfrac{1}{\sigma ...
- 199
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1
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69
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Proof that $Z = \frac{X}{Y}$ follows Cauchy distribution if X and Y are independent $N(0,1)$
In the book I am currently reading the statement in the title is proved by starting with:
$$F_Z(z) = Pr(X/Y < z) = Pr(X-zY, Y>0) + Pr(X-zY > 0, Y<0)$$
Then double integrals are used and ...
- 255
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1
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57
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Conceptual links between exponential decay and the normal distribution?
The standard normal distribution is given by
$$
f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}
$$
and exponential decay is given by
$$
N(t) = N_0 e^{-\lambda t}
$$
These formulae are equivalent if ...
- 11
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1
answer
31
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Variance of sum of differently weighted random variables with different variances and correlations?
I want to compute:
$Var[\sum_{i=1}^{n}w_iX_i]$, where $X_i\sim\mathcal{N}(\overline{X_i},\sigma_{i}^2)$ are joint normally distributed and $Cov[X_i,X_j]=\rho_{ij}\sigma_{i}\sigma_{j}$.
I would like to ...
- 49
1
vote
1
answer
39
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Conditional distribution on the linear inequality
A paper I am reading states the following Lemma as a well-known formula, but I can not find this formula anywhere else nor a complete proof of the lemma. I am also wondering if this lemma or something ...
- 13
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Finiteness of double integral involving conditional Gaussian density
Given that $\pi(W|V)$ and $\pi(W|V, Y)$ are both multivariate Gaussian distribution. $\pi(V|W)$ are some other proper density.
Does the finiteness of $$\int \int \pi(V|W) \pi(W|V) \, dW \, dV$$ ...
2
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1
answer
92
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Expectation of likelihood ratio involving Gaussian pdfs
I am interested in evaluating the following integral for a general value of $\mu \in \mathbb{R}$:
$$
I(\mu) = \int_{-\infty}^{\infty} \frac{2 e^{-0.5 (x - \mu)^2}}{e^{-0.5 (x - 1)^2} + e^{-0.5 (x + 1)^...
- 249
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51
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How to calculate the tail probability for multivariate normal distribution
Univariate Case: if $\sqrt{n}(\hat\mu-\mu)\overset{d}{\to}N(0,\sigma^2)$, then the tail probability can be approximated by
$$P\left(\hat{\mu}-\mu\ge t\right)\asymp e^{-nI(t)},$$
where $t\in\mathbb{R}$,...
- 701
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2
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51
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Standardizing the Equation of the Normal Distribution Curve
For context, I am a new educator teaching statistics.
The equation of any normal distribution curve with mean $\mu$, variance $\sigma^2$, and standard deviation $\sigma$ is
$$y=\frac{e^{\frac{-(x-\mu)^...
- 175
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3
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60
views
Suppose that one million people will vote in a primary with Candidate A vs. Candidate B. (Details below)
Suppose that one million people will vote in a primary with Candidate A vs. Candidate B.
a. Suppose that the one million people will vote for the two candidates by flipping a coin. So, Candidate A ...
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0
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30
views
Numerically stable way to compute probability between two points on a normal distribution
I want to compute the log probability between two points (x1, x2) on a normal distribution. The straightforward way is
log(CDF(x2) - CDF(x1))
but this is not ...
- 101
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0
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16
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Has the countable sum of random variables with gaussian tails again gaussian tails?
my question is if the countable sum of (positive) random variables with gaussian tails has again gaussian tails. More precisely I'm interested if the following sum
$\sum_{n=1}^{\infty}2^{-n}X_n$ has ...
0
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0
answers
42
views
Variance of linear combination of random variables conditional on another linear combination of the same random variables?
Suppose $X=\sum_{i=1}^nb_iX_i$ and $Y=w\sum_{i=1}^nX_i$. Is the following correct?
Conditional variance: $Var[X|Y]=\sum_{i=1}^nb_i^2Var[X_i|Y]+\sum_{i=1}^n\sum_{i\neq j}^nb_ib_jCov[X_i,X_j|Y],$
with:
$...
- 49
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187
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Concentration of measure for $X\cdot \mathrm{sign}(Y)- \mathbb E[X\cdot \mathrm {sign}(Y)]$ with $(X,Y)$ being multivariate Gaussian.
Say we have two $\mathbb R^n$ valued random vectors $X,Y$ such that $\left(\begin{array}\\X\\ Y\end{array}\right)$ is multivariate Gaussian with mean $0$. Given this one could apply similar methods as ...
- 329
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0
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13
views
Computing a Gaussian expectation
I am curious what the Bernstein-Orlicz norm of a (centered) Gaussian is. My Gaussians have pdf proportional to $\exp(-x^2/(2\sigma^2))$.
I thus need to compute
$$\tag{1}
\int_{-\infty}^\infty\exp(-x^2/...
- 7,114
1
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0
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40
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Integrate bivariate normal distribution over a polygon
Let $X\sim\mathcal{N}(0,I_2)$, with $I_2\in\mathbb{R}^{2\times 2}$ being the identity. I want to integrate its probability density function over a polygon $P$ whose (ordered) vertices are $\{p_1,\dots,...
0
votes
1
answer
50
views
Error propagation through rotations
Suppose you have a rigid body which goes through a sequence of rotations, each rotation is equal to a rotation matrix multiplied by some small random rotation parametrized by Euler Angles which are ...
- 635
1
vote
1
answer
26
views
Summation of any random variable and a gaussian random variable has a density function
I would like confirm if the below fact is true ask for a reference if it is true:
Let $X$ be a random variable, which does not necessarily have a density function. Let $G_{\sigma}$ be a gaussian ...
- 13
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0
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34
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Summation of two elements where the second one is recursively written as the sum of two other (sub)elements, then turn it into a random variable
Assume $x_r=x_1+x_2$, with $x_2=x_{21}+x_{22}$. Then, $x_r=x_1+x_{21}+x_{22}$. In turn, assume $x_{22}=x_{221}+x_{222}$, so $x_r=x_1+x_{21}+x_{221}+x_{222}$. Assume $x_{222}=x_{2221}+x_{2222}$. Then, $...
- 49
2
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0
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30
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Is a conditioned gaussian distribution sub-gaussian?
I am thinking that if $X_n$ converges to a gaussian distribution $N(0,1)$, whether $X_n|A_n$ converges to a sub-gaussian distribution, where $A_n$ is an event about $X_n$. Because we can show that
$$
\...
- 21
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0
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44
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weird cosine approximation comprising some weird sums and the theta function. why and how?
I randomly got recommended a video of some guy approximating the cosine function, and they eventually arrive at
$$\cos{\pi x}\approx\lim_{R\to\infty}\sum_{n=1}^{R}\left(\frac{-D^{-\left(x+1-2n\right)^{...
- 384
3
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0
answers
82
views
Derive a sharp bound of this integral using the transformation formula and/or Taylor's theorem
Let
$d\in\mathbb N$;
$\sigma\in\mathbb R^{d\times d}$ and $\Sigma:=\sigma\sigma^\ast$;
$b:\mathbb R^d\to\mathbb R^d$ be Borel measurable;
$\Delta t>0$;
$\kappa(x,\;\cdot\;)$ denote the normal ...
- 13.9k
2
votes
0
answers
71
views
Logarithmic transformation of Poisson R.V. is asymptotically normal as $\lambda\to\infty$?
Let $K\sim\operatorname{Poisson}(\lambda)$ and
$$
V(K)=\log(K/\beta+1)
$$
for some $\beta>0$.
Claim: The distribution of $V$ is asymptotically normal as $\lambda\to\infty$.
Thoughts:
Lemma #1: $K\...
- 6,279