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

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35 views

Sum of two normal distributions

Question: N(µ,σ) ξ ε N(3,4) ; η ε N(1,3) What is 3ξ + η? I'm new to normal distributions, and not even sure if this is called "sum of normal distributions". I ...
2
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2answers
60 views

Find the probability of getting 20 heads in 40 flips of a fair coin

Find the probability of getting $20$ heads in $40$ flips of a fair coin I did this problem with the binomial distribution and got my probability as $0.12537$. However, I am being asked to do this ...
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0answers
26 views

Can we take two temperature ranges for this probability?

"Electronic sensors of a certain type fail when they become too hot. The temperature at which a randomly chosen sensor fails is T ºC, where T is modelled as a normal random variable with a mean of ...
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0answers
95 views

Show that two linear combinations of i.i.d. normal random variables are independent

Let $\{ \vec{v}_1, \ldots, \vec{v}_n \}$ form an orthonormal basis for $\mathbb{R}^n$ and let $$ \vec{X} = \begin{pmatrix} X_1 \\ \vdots \\ X_n \end{pmatrix} $$ be a vector where each $X_i \sim \...
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2answers
64 views

Integrability of gaussian random variables

Let $(\Omega, \mathcal{F}, P)$ a probability field. Let $X : \Omega \to \mathbb{R}$ a gaussian distributed random variable. Show that $X \in L^p(\Omega, P)$, for every $p \geq 1$. Can someone, ...
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1answer
48 views

Normal Distribution Statistics [closed]

Hi I'm not so good in Statistics. and I'm trying to do as many examples possible. The question bellow is one of the example I was wondering if someone can help me understanding how to sole this sort ...
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1answer
62 views

Probability Distributions - Probability of x2 being larger than x1

I'm working with some probability distributions, working towards figuring the outcome of a statistical process. I have worked out my survey results, and I have a confidence interval (Confidence Level ...
3
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1answer
194 views

Normal approximation of tail probability in binomial distribution

From the Berry Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|\in O\left(\frac 1{\sqrt n}\right)$$ whereby $B_n$ has the standardized binomial distribution and $N$ has the ...
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1answer
128 views

Point of maximal error in the normal approximation of the binomial distribution

I am sorry for the long question! Thanks for taking the time reading the question and for your answers! Context: Let $B_n\sim\text{Binomial(n,p)}$ be the number of successes in $n$ Bernoulli trials ...
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1answer
29 views

Dependent Chi Square Distribution Random Variable

If $X, Y, Z$ are I.I.D normally distributed random variable. Then what is the probability density function for the function given by $\max(\mid x-y\mid ^2,\mid y-z\mid ^2)$.
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31 views

Characteristic function of $\chi^2$ distribution with one degree of freedom

Just for my own curiosity, I'm trying to derive the characteristic function of $X\sim\chi^2_1$, the $\chi^2$ distribution with 1 degree of freedom. According to wikipedia, it is $$ E[e^{itX}]=\frac{1}{...
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1answer
26 views

Where does mean and standard deviation go in the error function?

The error function is defined as $$ \textrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt~.$$ However, the normal distribution can take a more general form than the definition of the error ...
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0answers
24 views

Distribution of the modulus of multivariate normal distribution

Suppose we have a random variable X with density $$ f(x) := [2\pi \sigma]^{-n/2} \exp\left(-\frac{\|x-\mu\|^2}{2\sigma^2}\right) $$ where $x,\mu \in \mathbb{R}^n$ and $\|.\|$ denotes the euclidean ...
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0answers
43 views

Does a critical point of log-normal distribution exist?

To find a critical point(s) of a multi variable function we need to find the first partial derivative with respect to each variable, set each to zero and solve the system. We can then use the second ...
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0answers
86 views

Calculate new variance when sample size increases

Suppose I have 10,000 data points with the following statistics: Sample mean: 7 Sample variance: 4.2 (and hence standard deviation 2.05) Population mean is also 7. Now if I increase the sample ...
3
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2answers
200 views

Finding an error estimation for the De Moivre–Laplace theorem

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–...
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0answers
8 views

Is there any rationale behind the comparison of p values derived from the KS test?

I am trying to test normality for a system which contains 2D coordinate points. If they are close enough and if they have a roughly normal distance profile from their combined centroid, I would like ...
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1answer
34 views

What is wrong with my interpretation of a normal distribution question?

I am trying to solve the following testing problem, which can be rephrased as: A load of cargo contains 49 boxes where the weight of a box follows a distribution with mean µ = 205 pounds and standard ...
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0answers
42 views

Prove that $\Bbb P[\max_{k\le n}x_k<(2+)\sqrt{\log n}$ as $n\to \infty]=1$?

By the Borel-Cantelli lemma to prove that $\Bbb P[\max_{k\le n}x_k<(2+)\sqrt{\log n}$ as $n\to \infty]=1$ with a number $(2+)$ a shade more than 2. The hint is to use $(1-x)^n>1-nx$. Let ...
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1answer
30 views

Expected value of max(R-c,0)

Let $R$ bet Log-Normally distributed i.e. LN($\mu,\sigma^2$). Now we want to find the expected value: $\mathbf{E}[\max(R-c,0)]$ and also second order moment: $\mathbf{E}[\max(R-c,0)^2]$. I have ...
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1answer
28 views

Standard Deviation Adjustment

Is it true that to change the mean and stdev of a normal distribution (with mean 0 and stdev 1), all one has to do is to multiply the current PDF by the new stdev and add the mean? I.e. I have N(0,1),...
4
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1answer
89 views

Lower bound of Gaussian tail?

Have $N$ denote a $N(0, 1)$ random variable. I have found a $K_1$ such that for all $x >0$,$$\textbf{P}(N \ge x) \le K_1 x^{-1} e^{-x^2/2}.$$My question is, does there exist $K_2 > 0$ such that ...
3
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1answer
33 views

Which functions of a normally distributed random variable are also normally distributed

I know that if $X$ is normal then $Y$ = $f(X)$ = $aX + b$ is normal, and this is covered in other questions. Are there any other cases?
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15 views

Linear combination characterization of multivariate normal

Does anyone know a proof of the characterization of the multivariate normal as "X is multivariate normal iff every linear combination of its elements is distributed normally" ?
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173 views

Nearest neighbor for planar Poisson is normally distributed

Answering a recent question, I realized that the nearest point for a planar Poisson point process (with constant intensity $\lambda>0$) is normally distributed. Indeed, it is easy to see that if $...
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1answer
51 views

Is there a simpler way to calculate correlation?

Let's consider that a variable y constructed from x $x_i ∈ \left\{1,3,5,7,8\right\}$ $f(x_i)=2x_i+1$ $y_i=f(x_i) + ε_i, ∀i∈ \left\{1;...;5\right\} $ where $ε_i$ is a identically ...
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1answer
69 views

Distribution of $N(0,1)^2$

I'm trying to determine the distribution of $Z^2$ where $Z \sim N(0,1)$. I'm not really sure even how to start, is it ok to just multiply the density functions? May I need to use the MGF?
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0answers
72 views

Find the CDF of the sum of the inverse square of n random normal numbers

Question If I have n independent random normal numbers denoted $X_i$ each with mean $\mu_i$ and variance $\sigma_i$ (for $i = 1 ... n$). For each $X_i$ I have a weighting factor $w_i$. What is the ...
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0answers
43 views

Marginal distribution from conditional distribution

Let a random variable X be normal $N(\mu,\sigma^2)$ and let the conditional distribution of Y given X be normal $N(a_1+a_2X,\sigma_1^2)$. Find the marginal distribution of Y. I found their joint ...
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1answer
38 views

How to represent normal law and a variable intermingled?

Let's consider that a variable y constructed from x $x_i ∈ \left\{1;3;5;7;8\right\}$ $f(x_i)=2x_i+1$ $y_i=f(x_i) + ε_i, ∀i∈ \left\{1;...;5\right\} $ where $ε_i$ is a identically ...
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0answers
71 views

Density of sum of normal and cosine of a uniform random variables

Let X be a normal random variable with mean 0 and variance $\sigma^2$, let $\Theta$ be uniform on $(0,\pi)$, and let a be a real number. Assume X and $\Theta$ are independent. Find the density of Z=X+...
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1answer
33 views

Proving the $\operatorname{Var}(X)$ of $X\sim N(0,1^2) = 1$?

I've been struggling with this question for a while now. Question is on a normal distribution $ f(x)= \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} $, where the $E(X)$ was asked to be found, and hence ...
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1answer
25 views

Choosing the variance for a Gaussian Distribution for the Langevin Equation

I am trying to numerically integrate the Langevin equation which is given by $$ \ddot x= -\frac{6\pi\eta a}{m} \dot x + \frac{\xi(t)}{m} $$ where $\xi(t)$ is a probability distribution. I'm ...
2
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0answers
34 views

Proof of Conjugacy Between Multiple Multivariable Normal Distributions and Normal Inverse Wishart Distribution

I've been trying to prove that a normal inverse Wishart distribution can act as a conjugate to a series of multivariable normal distributions. Formally, $$\prod_{i=1}^I Norm_{\boldsymbol x_i}[\...
3
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1answer
22 views

Generating from $N_p(\mu,\Sigma)$ and Cholesky decomposition

I know how to generate random observation from $N_p(0,I)$ (applying Box-Muller transformation). But I was wondering how to simulate from $N_p(\mu,\Sigma)$(assuming $\Sigma$ to be pd). I started using ...
2
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1answer
73 views

What is the covariance of two dependent normal distributed random variables

Problem is about getting the covariance of two random variables that are not independent: $\operatorname{cov}(\tilde{x}\mid(\tilde{y}=y),\tilde{x})= \text{ ?}$ $$\tilde{x}\sim N(\mu_1,\sigma^2_1)$$ $...
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1answer
332 views

For the standard normal distribution N(0,1) , find the median, quartiles and interquartile range. (Give all answers to two decimal places.)

I know the mean is 0, which means the median is 0 and that the standard eviation is 1, but how do I find the quartiles? I know that the interquartile range is Q3-Q1. How would I begin solving a ...
2
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1answer
63 views

Does infinite repeated convolution with the same normal distribution converge?

According to Wikipedia, This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. ...
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1answer
102 views

Lognormal distribution function

What exactly is the Lognormal distribution? Also how can I find it's distribution. I came across the following problem in Sheldon M Ross, I am not understanding where to start. Please help A random ...
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0answers
30 views

How to fit a gaussian model on the overlapped area of two separate gaussians?

Suppose I have two 1D Gaussians distribution with N(U1,S1) and N(U2,S2) where U is the mean and S is the standard deviation. Suppose if we draw these two Gaussians, they overlap on a interval. Now ...
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0answers
22 views

How to estimate a normal distribution with mixture of gaussians?

I have a set of points which I can fit a Gaussian model on them using Maximum likelihood estimation. but this estimation is weak and I want to improve it. I want to fit a mixture of Gaussian on these ...
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1answer
32 views

Standard deviation misunderstanding

Here is a question that I stumbled upon (and the solution to it, my reasoning follows after the image): I answered "The two quantities are equal". My reasoning was as follows: the question mentions ...
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1answer
157 views

Bivariate normal distribution of points

I would like to generate points (x,y) in a 2-D plane that has a circular normal distribution similar to this: I found multiple terms for describing a "circular normal distribution" and yet, I'm not ...
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0answers
84 views

Normal and poissonian probability problems

I am working on a problem with a normal probability distribution but I am unsure of the results I calculated the probability asked for but still hesitate regarding the output and especially the first ...
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0answers
11 views

Finding Conditional Distribution of Multivariate distribution

Q: Supposing y ~ N4 (µ,∑) and µ = (1 2 3 -2)' ∑ =\begin{pmatrix} 4 & 2 & -1 & 2\\ 2&6&3&-2\\ -1&3&5&-4 \\ 2 &-2 &-...
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2answers
401 views

Normal Distribution: Probability of a Negative Value

The random variable $X$ can take negative and positive values. $X$ is distributed normally with mean $3$ and variance $4$. How can I find the probability that $X$ has a negative value?
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31 views

Proving that a subvector in a multivariate Normal distribution is also a Normal Distribution

Q: Assuming y = [y1 , y2, .. , yp]' is a p-dimensional random vector with mean vector µ and covariance matrix ∑. Given for any a = [a1, a2, .., ap]', we have a'y ~ N[a'µ, a'∑a]. Show ...
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1answer
33 views

Random variable with density function that is scaled geometric mean of density functions of two independent normally distributed random variables

Given two independent normally distributed random variables A and B: $$A \sim \mathcal{N(\mu_A, \Sigma_A)}$$$$B \sim \mathcal{N(\mu_B, \Sigma_B)}$$ is there a way to find normally distributed random ...
12
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1answer
104 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad T_N=\sum_{i=1}^{N}X_{i}^{2}\sim\chi^{2}\...
2
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1answer
57 views

Integrating a cost function over a normal distribution

Let's say you have a cost function $C(x)$ and you want to understand the expected cost if the input follows the normal distribution $$X \sim \mathcal{N}(\mu,\sigma ^2)\\ $$ If I want to find my ...