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

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73 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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1answer
99 views

On notation of $n$-dimensional Gaussian distribution

could anyone please help me with the following question? Let $\Sigma$ be the covariance matrix of an $n$-dimesnional Gaussian distribution. The generic requirement for $\Sigma$ is to be symmetric ...
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0answers
120 views

Multivariate Distribution & Bayes Rule

Suppose I have that an unknown vector, x, where x is drawn from the following distribution$ \bigl(\begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \bigr)$ ~ $N\bigl(0, \bigl[\begin{matrix} \sigma^2_1 ...
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1answer
807 views

Bayesian posterior with truncated normal prior

Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal{N}(\mu,\sigma^2)$. The variance $\sigma^2$ is known, $\mu$ isn't. We want to estimate ...
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390 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
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2answers
75 views

how to prove $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$

Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $N(0,\frac{a^3}{3})$. means $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
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1answer
65 views

If X is log-normally distributed prove the distribution function in terms of standard normal distribution?

I am not being able to solve part C and part D. Somebody please help! Thanks
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1answer
128 views

On the total weight of baseballs with normally distributed weights

Assume the weight (in ounces) of a major league baseball is a random variable, a carton contains 144 baseballs. Assume now that the weights of individual baseballs are independent and normally ...
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0answers
64 views

Conditional multivariate normal distribution

If $X = [X_1,\dots,X_n]$ is follows a multivariate normal distribution $\mathcal{N}(\mu,\Sigma)$, are there any (closed form) results known for the distribution of $[X_1,\dots,X_i \mid l_{i+1} < ...
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1answer
410 views

Sample from multivariate normal distribution with given positive-semidefinite covariance matrix

I want to draw a random vector from a multivariate normal distribution with given covariance matrix $Σ$. I'm following this algorithm: A widely used method for drawing a random vector $x$ from ...
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1answer
199 views

Probability with bullets and walls

There are two shooters with different guns and bullets. Each shooter shoots a bullet to a different target hanging on a wall. The hit of each bullet follows a normal distribution centered on its ...
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2answers
133 views
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1answer
35 views

normal as approximation to binomial

Among 784 checks, 479 had amounts with leading digits of 5, but checks issued in the normal course of honest transactions were expected to have 7.9% of the checks with amounts having leading digits of ...
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1answer
173 views

determine Fisher information of $N(0,\sigma^{2})$ over $\sigma^{2}$

So far i've got that $I(\sigma^{2}) = E_{\sigma^{2}}[\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(X)]^{2}$. And i got that $\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(x) = ...
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0answers
43 views

Mixture of Gaussians — Distribution Weight

I've been having trouble understanding how to simplify (as well as understand) the equation for what I'm calling the "Distribution Weight" of a Conditional Mixture of Gaussians distribution. Namely, ...
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2answers
112 views

How big of a sample size is necessary to be sufficiently confident in predictions?

A doctor at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be $90\%$ confident that her estimate is within $2$ ...
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1answer
46 views

Multiple independent random number streams

This question is somehow related to this one. Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution I want to do Monte Carlo simulations in ...
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2answers
96 views

Theoretical impossibility? Deviation from normality with a sample greater than 300?

Huge thanks in advance! I've been lead to believe that the following is a theoretical impossibility: a population larger than 300 records without an approximation of a normal distribution. The ...
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1answer
84 views

Generating 2D random vector from 4D covariance matrix

I have such covariance matrix $C$: ...
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0answers
78 views

Asymptotic (in)dependence of a maximum of an i.i.d. sequence of Gaussian random variables on a single random variable in this sequence

Suppose that I have a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$ where $X_i\sim\mathcal{N}(0,1)$. Denote the maximum of this sequence by $M_n=\max(X_1,\ldots,X_n)$. I ...
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1answer
97 views

Finding the variance of a normally-distributed random variable

X is a normally-distributed random variable, and $P[X<20] = 1/10 = P[X>100]$ I am trying to solve for the mean and the variance. I know that $\mu=60$ by symmetry. How can I solve for ...
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1answer
528 views

MLE of fourth moment of normal distribution

Take $X\sim N(0,\theta)$, and let $\phi = E(X^4)$, the fourth moment. What is its MLE, $\hat{\phi}$, and what is the asymptotic distribution of $\sqrt{n}(\hat{\phi} - \phi) $ as $n\to \infty$? Any ...
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1answer
106 views

Finding the expectation of functions of random variables with a bivariate normal distribution

X and Y have a bivariate normal distribution. I am given that $E[X] = 4$ and $E[Y] = 10$. I am asked to find $E[X^2 - Y^2]$ WITHOUT integration. I know how to solve for this using integration, but ...
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3answers
89 views

$Z$ score probability

I was given a question where I was supposed to find the probability of obtaining $y$ between two scores, however when I input my answer it tells me that I'm wrong, the question is given below along ...
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2answers
370 views

Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables

It's well known that, for a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$, where $X_\max=\max(X_1,\ldots,X_n)$, the following convergence result holds: ...
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1answer
282 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
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0answers
172 views

Fourier transform of a complex exponential with quadratic argument

I'm a PhD student who is starting to work right now in the well-established field of ultra-fast optics. The thing is that, in most of the papers I have been reading during the past few days, there is ...
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1answer
68 views

Bigger than and equals rewritten in normal distribution question

So it is correct to say that $P(482\le x \le 510) = P(x \le 510) - P(x < 482)$ where x is a random variable in a normal distribution? Thanks!
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2answers
167 views

Calculating integral with standard normal distribution.

I have a problem to solving this, Because I think that for solving this problem, I need to calculate cdf of standard normal distribution and plug Y value and calculate. However, at the bottom I ...
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1answer
38 views

Probability , Geometric and Gaussian

So,I'm good at the questions which require the understanding of basic formulaes , but this one my prof said needs me to think (for the first one)'geometrically'=Stumped. Please Help! The second is an ...
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0answers
134 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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1answer
65 views

normal distribution expected value

In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html $$f(x) = \sqrt{\frac{k}{2\pi}}e^{-\frac{k(x-\mu)^2}{2}}$$ they let $x-\mu = v$ $dx = dv$ and conclude that ...
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2answers
205 views

Norm of random vector plus constant

Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x $$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$ but I cannot prove it. We ...
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1answer
3k views

How to calculate the probability of a normal distribution with unknown mean and unknown variance?

How do you calculate the probability of a normal distribution with unknown mean and unknown variance? If a problem stated, for example, that 15% of the time sales are more than 15,000 and 20% of the ...
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1answer
186 views

Calculating an average on normal distribution

Given the fair dice, if the result is $1$ or $2$ the profit is $3$USD, if the result is $6$ you don't win or lose anything, for every other result you lose $2$USD. What is the average profit, that ...
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1answer
29 views

In statistics, what is the meaning of $Z_{0.3}$

What is the meaning of $Z_{0.3}$ and how do I calculate it? I know it was calculated this way: $$Z_{0.3} = -Z_{0.7} = -0.52$$ I tried to follow the General Distribution table but I can't seem to ...
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1answer
587 views

Normal distribution, chi-square distribution and t distribution combiened

How to prove that when X is from Normal Distribution and Y is from Chi-square Distribution with parameter f and X,Y are independent then X/sqrt(Y/f) is from t distribution with parameter t? I got ...
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2answers
274 views

Conditional mean and variance of normal random variables

There are two independent normal random variables $N_1, N_2$ with means $\mu_1, \mu_2$ and variances $\sigma_1^2, \sigma_2^2$ respectively. Is there a way to compute the two conditional expressions ...
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1answer
68 views

What is the effect of the variance on a sequence of cumulative product?

We randomly draw numbers from a normal distribution with mean equals $mu$ and variance equals $var$. We draw the values: $x_1, x_2, x_3, x_4, ...$ Then, we construct a sequence made of the ...
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1answer
88 views

Can the characteristic function of a multivariate normal distribution be extended from a neighborhood of the origin?

Let $x$ be a scalar random variable. There is a theorem that states that if $E[\exp(ixs)]= \exp\Big( i{s}\mu - \tfrac{1}{2} {\sigma^2s^2} \Big)$ for some neighborhood around the origin (i.e. ...
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0answers
109 views

Summing many non-standard i.i.d. uniform random variables

all! I have looked up a fair bit on this question and learned much about the problem. But haven't been able to get any crisp answers. Sorry, if I'm missing something obvious. I know one can use the ...
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2answers
117 views

Multivariate normal distribution from invertable covariance matrix

I want to generate a random vector with $\mathcal{N}(0, C)$ distribution, i.e. normal distribution with $0$ mean and given covariance matrix $C$. $C$ is not invertible (singular). Here it's written: ...
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1answer
97 views

The MLE of a $N(\theta, 1)$ distribution

I am trying to find the Maximum Likelihood Estimator of an i.i.d. sample $X_1, \ldots, X_n$ arising from the model $N(\theta, 1)$, where $\theta \in [0,\infty)$. I have done this problem previously ...
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0answers
109 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
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2answers
89 views

Bound of Standard Normal Integral

Consider the Standard Normal Integral given by: $$ I=\int_{-\infty}^{\infty} \frac{1} { \sqrt{2\pi} } e^{ \left( -z^2 /2 \right)} dz $$ In order to prove that it exists we note that the integrand is ...
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1answer
114 views

The probability that a randomly chosen grain weighs less than the mean grain weight

If Y has a log-normal distribution with parameters $\mu$ and $\sigma^2$, it can be shown that $E(Y)=e^\frac{\mu + \sigma^2}{2}$ and $V(Y)=e^{2\mu +\sigma^2}(e^{\sigma^2}-1)$. The grains composing ...
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2answers
47 views

Why are vectors $X_2$ and $X_3$ bivariate normally distributed?

I have a stochastic vector $\mu = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}$ and $\Sigma= \begin{bmatrix}1 & 0 & -1\\0 & 2 & 0 \\ -1 & 0 & 3\end{bmatrix}$. I have to proof that ...
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1answer
522 views

Exponential and Uniform distribution with conditional probability

A computer lab has two printers. Printer I handles 40% of all the jobs. its printing time is Exponential with the mean of 2 minutes. Printer II handles the remaining 60% of jobs. Its printing time is ...
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1answer
176 views

Forumla for finding conditional variance

I need to find the conditional variance $\mathop{\mathrm{Var}}(X_1|(X_2+X_3))$, given that $X_1\sim N(0,1)$ and $X_2+X_3\sim N(0,2+2\gamma)$. The covariance between X1, X2+X3 is $\rho$. From this ...
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1answer
37 views

Calculating probability of difference of two distributions.

A has normal distribution of scores of students with $X \sim \mathcal N(625, 100)$ B has normal distribution of scores of students with $X \sim \mathcal N(600, 150)$ Now I have to calculate ...