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

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8
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2answers
125 views

Multivariate Normal Distribution: Relationship between two conditional probabilities.

Suppose I have a multivariate normal random variable $Z$ which has $n$ dimensions. Suppose I have a vector $x$. Set $i$ as a number between $1$ and $n$ and $k$ as a number between $1$ and $n-1$. Can ...
1
vote
0answers
57 views

If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
2
votes
1answer
86 views

Find all solutions of $f$ such that $ \operatorname{Cov}(f(x),x)=c $ where $x \sim N(\mu,\sigma^2)$ and $c$ does not depend on $(\mu,\sigma^2)$

Find all solutions of $f$ such that $$ \operatorname{Cov}(f(x),x)=c$$ where $x \sim N(\mu,\sigma^2)$ and $c$ does not depend on $(\mu,\sigma^2)$ I think the only solution is $f$ is constant (almost ...
1
vote
1answer
47 views

Normal Distribution Application

Given: $\mu=80$, $\sigma=15$, $500$ respondents a. Find $P(74\lt x \lt 101)$ b. Find number of respondents with score $\lt 98$ For a., my answer is $0.5746$ (using the formula $z=\cfrac{x-\mu}{\...
1
vote
0answers
35 views

How to sample multivariate random normals?

Suppose $\Omega=(0,1)$, $\mathcal{B}$ is the Borel $\sigma$-algebra on $(0,1)$ and $P$ the Lebesgue measure. Let $\Phi:\mathbb{R}\to(0,1)$ denote the standard normal CDF. Then, we can generate $X\sim ...
0
votes
0answers
60 views

What do we know about the sum of two dependent lognormal random variables (without assuming underlying joint normality)?

Let $V\sim\mathcal N(0,\sigma^2)$ and $W\sim\mathcal N(0,\zeta^2)$ marginally, but without any assumption of being jointly normal. For given constants $a$ and $b$ (real, or maybe restricted to ...
0
votes
0answers
10 views

distributions with kurtosis

I have a stream of events from a large set, S. I am counting their residues modulo c in a set of counters. (c is much smaller than the cardinality of S.) If all of the S events were equally probable,...
0
votes
1answer
22 views

Create Normal Distribution Curve for Data

Following this tutorial I would like to plot normal distribution curve for following data ...
1
vote
1answer
52 views

Normal distribution with P(x=a) and P(x≥a)

Given: height of $1000$ students normally distributed with $\mu=174.5\,\mathrm{cm}$, $\sigma=6.9\,\mathrm{cm}$ Find: a. $P(x<160\,\mathrm{cm})$ b. $P(x=175\,\mathrm{cm})$ c. $P(x\geq188\,\...
0
votes
0answers
23 views

Chi-Squared distribution

For $X = [X_1, X_2, \ldots, X_k] = N_k(\mu, \Sigma)$ and $\mu = \mu\boldsymbol{1}$ for a constant $\mu\in\mathbb{R}$ and $\Sigma = \sigma^2I_k$, let $\bar{X} = \boldsymbol{1}^TX/k$ and $S^2 = (X-\bar{...
0
votes
1answer
36 views

Evaluation of the integral of a linear gaussian model?

suppose the following $$P(x_1) \text{ ~ } N(7.3555\ ; 12.3433)$$ and $$P(x_2|x_1) \text{ ~ } N(0.995\ + \ 0.2351x_1\ ;13.732)$$ how to evaluate the following integral? \begin{align*} \int \mathcal{I}...
1
vote
0answers
23 views

Covariance estimation and Graphical Modelling

I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions. Suppose, $X_i \overset{iid}{\sim} N_p(\mu,\Sigma)$, $i=1(1)n$...
0
votes
0answers
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
votes
2answers
61 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 ...
0
votes
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 ...
0
votes
0answers
96 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 \...
0
votes
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, ...
-2
votes
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 ...
0
votes
1answer
63 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
votes
1answer
198 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 ...
4
votes
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 ...
0
votes
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 $\max(\mid x-y\mid ^2,\mid y-z\mid ^2)$.
1
vote
0answers
32 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}{...
1
vote
1answer
27 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 ...
0
votes
0answers
25 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 ...
0
votes
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 ...
0
votes
0answers
88 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
votes
2answers
209 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–...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
31 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
votes
1answer
90 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
votes
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?
0
votes
0answers
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" ?
2
votes
0answers
180 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 $...
1
vote
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 ...
1
vote
1answer
70 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?
1
vote
0answers
75 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
74 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+...
0
votes
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 ...
1
vote
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
votes
0answers
35 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
votes
1answer
23 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
votes
1answer
76 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: $\operatorname{cov}(\tilde{x}\mid(\tilde{y}=y),\tilde{x})= \text{ ?}$ $$\tilde{x}\sim N(\mu_1,\sigma^2_1)$$ $...
0
votes
1answer
337 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
votes
1answer
67 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. ...