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

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0
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1answer
18 views

Correlate normal shocks

I am trying to generate some random standard normal variables and correlate them In particular I want: $$ \bf Y \sim \mathcal N(0, \Sigma) $$ where $\textbf{Y} = (Y_1,\dots,Y_n)$ is the vector I ...
0
votes
1answer
51 views

Variance of squared random variable

Can anyone help to prove this equation for any distribution $$ E(z^4)=1+\operatorname{Var}(z^2) $$ where $z$ is a random variable with the standard normal distribution $$z=\frac{x−μ}σ$$
0
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1answer
21 views

Unconditional Variance of Normal RV with mean being a NRV

I am trying to find the variance of $X$ which is defined like this: $$X \sim N(Y,e)$$ where $Y$ is a normal random variable with the distribution $Y \sim N(a,b)$. $a$,$b$, and $e$ are known ...
3
votes
2answers
56 views

Log - Normal Distribution

could someone explain why the log-normal distribution's mean is $$ e^{u + {\sigma^2\over2} } $$ and the variance is $$ (e^{\sigma^2} -1)e^{{2u} + \sigma^2} $$ I'm not too sure how ...
0
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0answers
18 views

Multivariate probit gaussian convolution

For univariate normal distribution, we know the following formula exists $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Is there a similar formula for ...
1
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1answer
64 views

Compound Distribution — Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
1
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1answer
68 views

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}$dx?

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $$\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}dx?$$ I just need to solve this integral. But, I don't know how. ...
1
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0answers
23 views

Is the following function symmetric?

I am reading the paper Hierarchical Clustering of a Mixture Model by Goldberger et al. On the page 2 of this paper they define the following function: $ d\big(\mathcal N(\mathbb\mu_1, ...
0
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1answer
46 views

“Exact” probability from Normal distribution

I have a small problem. Lets say I have a system that emits random distances from a certain point in space, its a real number from a normal distribution with mean = 0 and a set variance I know (lets ...
0
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2answers
27 views

Is the joint distribution of these two dependent Gaussian RVs, Gaussian?

I have two dependent Gaussian variables $X_1,X_2$ with unit mean each, and standard deviations $\sigma_1=2a$ and $\sigma_2=a$ respectively, while $a>0$ Is the joint distribution of these two ...
0
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0answers
36 views

Joint Distribution of Jointly Gaussian Variables

Random variables $X$ and $Y$ are jointly Gaussian where $E(X)=a, Var(X)=b, E(Y)=c, Var(Y)=d, Cov(X,Y)=e$. How can I find the joint distribution of $X$ and $Y$?
1
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1answer
54 views

Let $X$ and $Y$ be of the same dimension and jointly normal. Find the distribution of $X+Y$.

Let $X$ and $Y$ be of the same dimension and jointly normal. Find the distribution of $X+Y$. Can we start off by saying that if $X$ and $Y$ are jointly normal, then $X$ and $Y$ are normal as well?, ...
0
votes
1answer
49 views

Why is $X/\|X\|_2$ uniformly distributed on a unit sphere when X is n-dimensional standard gaussian vector?

In the proving the above, I see that since $X$ is multivariate gaussian then for any orthogonal matrix $Q$ we have that $QX$ is standard multivariate gaussian. Then I somehow reasoned that ...
1
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2answers
46 views

Normal approximation of a binomial distribution

A car assembly line produces 1920 cars per shift. A defect rate of 3% is considered acceptable. From the production of one recent shift, 65 cars were found to be defective. What is the probability of ...
0
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2answers
28 views

What does one sample average tell me about the average of other samples?

I'm sure this has been asked before but I can't find the exact variant I'm looking for. I have an infinite population of elements. I take a sample of those, say 10000, and make a measurement for ...
0
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0answers
26 views

Distribution - magnitude of vector of 2 orthogonal Gaussians with unequal variance (generalisation of Rayleigh distribution)

A Rayleigh distribution results from the magnitude of a vector of 2 orthogonal Gaussians with equal variance and zero mean, however if the equal variance assumption is relaxed, what distribution ...
0
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0answers
51 views

find limit distribution by using central limit theorem.

$$x_1,...,x_n \sim \text{uniform (0,1)}$$ $$Y_n=\sum_i^n X_i$$ I want to find limit distribution by using central limit theorem. $E(Y_n)=n/2$ and $V(Y_n)=n/12$ And Moment generating function ...
0
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0answers
16 views

Limiting distribution of the barycenter $\nu_n=\mathcal{N}\Big(\frac{1}{n}\sum_{i=1}^{n}m_i, (\frac{1}{n}\sum_{i=1}^{n}\sigma_{i})^2 \Big)$

I have a set of gaussian measures $\mu_1, \dots, \mu_n$ with $\mu_k=\mathcal{N}(m_k, \sigma_k)$. I am interested in the empirical Wasserstein Barycenter of this sequence which corresponds to ...
0
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1answer
16 views

Proof regarding standard normal distribution

I am struggling to prove the following: If Z~N $(0,1)$, prove that for the positive k, $P(|Z|<k)=2-2 \Phi (k)$ I know that $P(|Z|<k)$ can be written as $P(-k<Z<k)$ and that $\Phi (k)$ ...
0
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1answer
60 views

Ratios of median/mean and standard deviation/IQR in a normal distribution

I have some queries on the following question For a normal distribution, find the ratios of: (a) $\frac {\mbox{median}}{\mbox{mean}}$ (b) $\frac {\mbox{standard deviation}}{\mbox{interquartile ...
1
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0answers
73 views

Theoretical distribution of a random variable

Martin has $n$ words, and he wants to make a computer program that chooses for him $k$ words (and shows them to him), where $k \le n$, for as many times as he clicks a button until all of the words ...
0
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1answer
28 views

Normal distribution question involving absolute value

I don't quite understand how to do the following question. How I tried to do it is to imagine the normal distribution curve, with the highest peak at 4. I understand that |Q| means the absolute value ...
0
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1answer
30 views

Normal Distribution, percent less than mean minus one std. dev

GRE Question, part b. My answer is 16%, but the book says 15%. Why? mean = 65 cm stdev = 5 cm Probability that penguin chosen at random will be less than 60 cm is 16%, it shows in Normal ...
1
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1answer
55 views

Compute the probability of a joint event involving two independent standard normals

Suppose $X$ and $Y$ are independent, standard normal random variables. I'm trying to compute the probability of the event $$ \{X \leq x, Y \leq kX\} $$ where $k$ is a positive constant. The ...
0
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0answers
40 views

A univariate normal distribution with zero variance is a point mass on its mean

On Wikipedia, that's what appears in the first bullet point of the 'definition' section. What's a point mass? I've tried googling it although all I've found is a definition of 'point mass' from a ...
0
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0answers
22 views

Why Gauss distribution is a necessary condition for the equality of maximum likelihood estimate and sample mean

I'm reading Probability Theory: The Logic of Science by Jaynes. And while I'm reading section 7.3, he proofed that Gauss distribution is a necessary condition for the equality of maximum likelihood ...
3
votes
1answer
51 views

How can I draw the skewed normal distribution curve

If its of real importance, I am trying to plot the data on gnuplot. I have the following of some experimental data, obtained by ...
1
vote
1answer
35 views

Distribution of $X\cdot Y +a\cdot X$ for $X,Y$ standardnormal

I am searching for the exact or asymptotic CDF of the rv $X\cdot Y +a\cdot X$ with the $X,Y$ independent standard normal rv's. Found nothing till now. Any hints? Thanks.
1
vote
1answer
26 views

Correlation and Covariance

The book I'm reading gives this as an example for lognormal variables. Starting at some fixed time, let $S(n)$ denote the price of a security at the end of $n$ additional weeks, $n \ge 1$. A popular ...
2
votes
1answer
60 views

Two step random experiment: Density of combined uniform and normal distribution

imagine a random experiment, where first some number $u$ is drawn uniformly on $[c-\varepsilon,c+\varepsilon]$ for $c>0$ and $0<\varepsilon<c$. Next, a $N(u,\sigma^2)$-distributed random ...
2
votes
2answers
59 views

Why is the integral $\int_0^1t\,dW_t$ a normal random variable?

Consider the random variable $X=\int_0^1t\,dW_t$, where $W_t$ is a Wiener process. The expectation and variance of $X$ are $$E[X]=E\left[\int_0^1t\,dW_t\right]=0,$$ and $$ ...
1
vote
1answer
20 views

Check if $Cov(X_1+X_2,X_1-X_2)=0$, i.e. if independent?

"Let $X_1$ and $X_2$ be independent, $N(0,1)$-distributed random variables. Show that $X_1+X_2$ and $X_1-X_2$ are independent." I know that for multivariate normal distributions independence can be ...
1
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0answers
34 views

expectation of normal-wishart distribution

I want to compute $ E[\mu\Lambda] $ for a normal-wishart distribution how can i compute it? A normal-wishart distribution is defined as below: $$ ...
2
votes
1answer
38 views

Hypothesis testing (parameters)

I am new to statistics, but familiar with basic concepts. The lecture that confuses me is Hypothesis testing. Although the idea behind it seems ok, I couldn't understand the first exercise I found. ...
1
vote
1answer
51 views

Mean of exponential Brownian motion

I am new to stochastics and I am trying to compute the expectation of $S_t = e^{\sigma W_t}$, where $W_t$ is a standard Brownian motion and $\sigma>0$. My attempt (using the log-normal PDF here and ...
2
votes
1answer
31 views

Distribution of $X$, when distribution of $\ln(X/X_0)$ is normal?

We assume that $\ln{(S_T/S_0)} \sim N(\mu T, \sigma^2 T)$. I have in suggested solution that $$\ln(S_T/S_0) = \mu T + \sigma \sqrt{T} Z$$ where $Z \sim N(0,1)$ or $S_T = S_0e^{\mu T + \sigma ...
2
votes
1answer
48 views

Conditions for convergence and the multivariate central limit theorem

Consider $r$ independent but not identically distributed random vectors $x_1,\dots,x_r$, each of dimension $d$. The elements of $x_i$ are integer valued elements in the range $-d,\dots,d$ with mean ...
1
vote
1answer
27 views

Finding the probability of X_Bar with sample variance included?

The question I am asked is $P(\bar{X} > 3 + 0.4984S)$, where I am additionally provided $n = 25, \mu = 3.0, \sigma^2_\text{pop} = 3.0$. $\bar{X}$ is the sample mean and $S$ is the sample variance. ...
3
votes
0answers
42 views

binomial distribution converges in distrubtion to standard normal distribution [duplicate]

$$x\sim \frac{x-np}{\sqrt{npq}} \overset{d}{\to} N(0,1)$$ I want to show that normalised binomial distribution converges in distribution to standard normal distribution. Note that: Convergence in ...
0
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0answers
13 views

Simplifying an expression which includes summation symbols and the cumulative distribution function for the normal

I would like to be able to simplify the expression: $E(Y|\mu,\sigma^2) = \frac{\sum_1^J 1 - 2 \Phi((c_j - \mu)/\sigma) + 2 \Phi((c_j - \mu)/\sigma)^2}{J}$ where $\Phi$ is the cumulative distribution ...
1
vote
1answer
55 views

From normal to normal standard distribution + a bonus Q

I am failing to see where the $\sigma$ is going in the below. Given that normal distribution's pdf is: $$p(x) = \frac{1}{\sigma \sqrt{2 \pi}}\exp \left( -\frac{(x-\mu)^2}{2 \sigma^2} \right)$$ and ...
0
votes
1answer
43 views

What is the analogue of $f(x)=e^{-x^2}$ on the torus? What about its Fourier transform?

Let $f:\mathbb R \to \mathbb R$ such that $f(x)=e^{-x^2} \ (x\in \mathbb R).$ We know that $f, \hat{f} \in L^{1}(\mathbb R).$ My Question is: What is the natural analogue function of ...
4
votes
1answer
122 views

Can a class test scores with a bimodal distribution provide statistical evidence for cheating?

I know the normal distribution can represent many things in nature. Most items are normally distributed. I recently watched a video of a professor who claims that biomodal distributions provide ...
1
vote
1answer
77 views

Variance of a normal distribution for coin toss.

I have difficulties constructing the normal distribution for (20) coin tosses. (Don't ask why, but I never had probability in school.) What is the probability of getting at most 12 heads out of 20 ...
1
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0answers
23 views

Finding conditional probability distribution (X|Y) from (Y|X)

$(Y,X)$ has a joint distribution where the marginal of $X$ is a standard normal and $Y|X \sim U \left[|X|-\frac{1}{2},|X|+\frac{1}{2}\right] $ where $U[a,b] $ means uniform in the interval [a,b]. How ...
6
votes
2answers
92 views

Recurrence for expected length of Gaussian vector

Let $g_k \sim N(0, I_{k \times k})$ be a a standard $k$-dimensional Gaussian vector. Denote by $\|g\|$ the $2$-norm of $g$. By explicit integration, it is not hard to see that $$ \mathbb E \|g_k\| = ...
0
votes
0answers
12 views

Properties of $tk$ in confidence intervals for the mean of a normal distribution

I have a question about $tk$ in confidence intervals for the mean of a normal distribution. Where $t_1- \frac{a}{2}$, $n - 1$ is a quantile of the level $1- \frac{a}{2}$ for $t$ Student distribution ...
1
vote
1answer
33 views

Family of functions with specific properties

I am wondering if there is possibly a well-known family of functions $f_l:\mathcal{R}\to\mathcal{R}$ parametrized by a single positive real / integer value $l$ that has the following properties: 1). ...
1
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0answers
20 views

Bivariate normal: normal difference distribution

I know that the difference of two multivariate or univariate normally distributed random variables produces another (multivariate) normal random variable. But, what happens when you take the ...
2
votes
0answers
50 views

Distribution of $aX+bX^2+cX^3$ where $X$ is standard normal

I am looking for some distributional characteristic (for example a characteristic function) of a random variable which is defined as $aX+bX^2+cX^3$, where $X$ is a standard normal variable. Is there ...