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

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2
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3answers
55 views

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian. Mathematically, if we know that $a_1X+b_1Y$ and $a_2X+b_2Y$ have Gaussian ...
0
votes
0answers
28 views

Mixture of binomial distributions

I have a population of agents with a single real-valued attribute $x$. Each of them performs $n$ Bernoulli trials with success probability $q(x)$ which depends on their attribute. In particular, $$ ...
0
votes
1answer
15 views

Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
1
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1answer
29 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
1
vote
0answers
20 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
2
votes
1answer
60 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the symmetric semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes ...
2
votes
1answer
41 views

Are $X$ and $Y$ necessarily normal if the the sum $Z=X+Y$ is normal?

Of course, asking the question the other way round is straightforward to answer as via the convolution we find that the sum of two normal distributed variables is again normal. But however, is it ...
1
vote
2answers
39 views

Normal Distribution of Sums

I have two normally distributed random variables $X$ and $Y$. Then I know that the sum $X-Y$ is also normally distributed (i). However, I want to show (preferably by a counter example) that the ...
0
votes
0answers
38 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...
0
votes
2answers
27 views

What is a decision threshold and how does it apply to a statistical power?

I'm pretty confused on what is actually going on in this section with hypothesis testing. As another note, the values below are computed using R. I have a homework problem that says: From the ...
0
votes
0answers
9 views

convergence multivariate normal

If $X$ and $Y$ have asymptotic normal distribution then using Slutsky's theorem $aX+bY$ is also asymptotic normal, can I conclude that the vector $(X,Y)$ is asymptotic bivariate normal? If not, how ...
6
votes
1answer
97 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
0
votes
1answer
42 views

chi square distribution probability

I am having a problem with this. Suppose a stock's returns are normally distributed with mean $m$ and variance $\alpha^2$ and we compute the sample variance from a sample of $41$ periods and find ...
4
votes
1answer
71 views

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
1
vote
0answers
22 views

Distribution of unknown covariance matrix, given variance of linear combination

Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$? Specifically ...
1
vote
1answer
14 views

When the domain of a continuous distribution exceeds feasible values, what should I do?

Now I need a (maybe approximated) model for this distribution: $$X=(x_1, x_2, …, x_n)$$ where $x_i$ is a real number between $0.0$ and $1.0$, and the sum of $x_i$ equals $1.0$. Now, I want to use ...
1
vote
0answers
26 views

MLE of variance for a spherical Gaussian

I am trying to implement the X-Means clustering algorithm. In it, the authors use the BIC to determine which model fits the data best. It is explained here: ...
1
vote
2answers
111 views

Why does normal distribution have the same shape regardless of its parameters? [closed]

The formula for normal distribution is quite complicated, it has $\sigma$ in the exponent and in denominator. And no matter what $\sigma$ is, the shape of its pdf is the same (i.e. for example 3 ...
1
vote
2answers
21 views

Formula to get mean and standard deviation of this multi-variable equation

$$ \binom n x \times\left(\frac1r\right)^x\times\left(\frac{r-1}r\right)^{n-x} $$ If you have $n$ boxes and have a $\frac1r$ chance to fill each one, this equation returns the chance that you fill ...
1
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0answers
29 views

How good of an approximation is a normal probability distribution for sum of dice rolls?

I want to know how well the normal distribution explains the sum of rolls with n dice with s sides. The mean value and the variance of the dice rolls are $$\mu=n\frac{s+1}{2}$$ and ...
1
vote
1answer
25 views

Approximation of distributions with dice

I want to know what dice to roll to get a given probability distribution(mainly normal distributions but exponential distribution would also be helpful). I want a function $f$ so that ...
1
vote
1answer
44 views

Calculating probability of a normal distribution, not getting correct answer

I'm doing a homework assignment and having some trouble matching the correct answers from my professor. As a reference, I'm calculating these answers using R. The question is as follows: Assuming ...
-3
votes
2answers
53 views

Black–Scholes but probably basic stats [closed]

Hello friends! I'm rusty (bad) with my statistics and this problem is bugging me, so any help would be greatly appreciated! Just really bad at figuring out how the 1-N() gets transformed into the ...
1
vote
0answers
14 views

Does Bivariate Normal have an MLR?

In general, with all parameters unknown I think the answer to this question is no. I think this because in this instance we would have a curved multivariate exponential family. Is this reasoning ...
0
votes
0answers
26 views

Gaussian vector multiplied with a matrix is another Gaussian vector: How to show?

Assume that $w$ is a $M$ dimensional random vector, such that: $w \sim N(w|0,\alpha^{-1} I)$. Now I have a $N \times M$ matrix $\Phi$, which is not random. I want show that the vector $Y= \Phi w$ is ...
0
votes
1answer
30 views

Sum of gamma and normal random variable

If $X$ has an exponential distribution and $Y$ is normally distributed random variable, then what is the distribution of $Z=X+Y$?
0
votes
1answer
33 views

Gaussian integral justification

So I read on the very nice proof of the Gaussian integral being equal to the root of pi and its application for the normal distribution (in fact the normal distribution is described by the Gaussian ...
0
votes
1answer
56 views

How to calculate a population mean for a normal distribution

This is for homework, but I'm a bit confused on how I can find $E(X_i) = \mu$ given a normal distribution. The question is as follows: In a farm, let $X$ denote the number of fruits harvested in a ...
1
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0answers
44 views

Series of sums of normal variables, likelihood principle

Suppose I have a series of normal variables $Y_i \sim \mathcal N(\theta, 1)$ for $1 \leq i \leq N$. Define: $$S_k = \sum\limits_{i=1}^kY_i$$ Since they're sums of normally distributed variables, ...
1
vote
1answer
14 views

Scaling Normal Distribution

Why is it that $N(0, ct) = \sqrt c N(0,t)$? What does it mean when we take a constant out of a distribution?
1
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0answers
53 views

Find $E[X+2Y|Z]$

$X,Y$ are independent standard normal. Let $W=X+Y$, $Z=X-Y$. Find $E[X+2Y|Z]$ Attempt: $E[X+2Y|Z=z] = E[X+2Y|X-Y=z] = E[Y+z+2Y] = 3E[Y]+Z = Z$ Is this correct?
0
votes
0answers
29 views

Independence of a linear combination of Normal Random Variables

I would like to prove the following: I have that $X_1, X_2$ are 2 random variables, each independently following a $N(\theta,1)$ distribution. I firstly need to show that: $ X_1 - \bar{X}$ is ...
2
votes
1answer
43 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
0
votes
0answers
18 views

Use of normal distrubution to determine price differences

Currently I am analyzing price data of many of our competitors. For example: Company A sells product X for $45 Company B sells product X for $44 Company C sells product X for $52 We sell product ...
1
vote
1answer
46 views

Integrate bivariate normal distribution over circular region

Context: Need to compute the probability that a 2D Gaussian random walk falls within distance $ d $ of some point $ p $ on the next step. (Assume the covariance $ \Sigma $ is the identity matrix $ I ...
0
votes
0answers
24 views

Expected values from normal distribution

I'm stuck on a statistics question from my university past papers. The question is: The bit I'm stuck on is calculating the expected values from the truncated normal distribution So I'm really ...
3
votes
2answers
33 views

Normal approximation of binomial distribution - limits

In binomial distribution number of successes (usually denoted as $x$) must be between between $0$ and $n$, inclusive ($n$ is the number of trials). So for example there can be a problem which asks for ...
1
vote
1answer
58 views

Integral of multplication of normal pdf and Rayleigh pdf distribution

I need to calculate following definite integral $$\frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx$$ It is actually ...
1
vote
1answer
36 views

Distribution of the sum of squared independent normal random variables

How do I go about finding the the pdf of the statisitc $\sum_i x_i ^2$ such that each $x_i$ is iid from a $N(\sigma , \sigma)$ distribution? I've searched, but cannot find a straightforward answer. ...
0
votes
1answer
24 views

Distribution of Logistic of Normal

If $X \sim N(\mu_X, \sigma^2_X)$ and $Y= \frac{\exp(X)}{1+\exp(X)} $, what is the distribution of $Y$? I thought logit-normal would fit the bill, however the logit of $Y$ is ...
1
vote
0answers
32 views

distribution of quadratic form of jointly normal random variables?

I need to derive the distribution of the random variable $\frac{W'(I-1(1'1)^{-1}1')ZZ'(I-1(1'1)^{-1}1')W} {Z'(I-1(1'1)^{-1}1')Z}$ , where $(Z, W)'$ ~ $N(0, I), \,Z=(Z_{1}, ..., Z_{n}), \,W=(W_{1}, ...
1
vote
1answer
61 views

Integral involving CDF of a normal distribution

Can we evaluate the following integral ? $$\int_0^\infty x e^{-x^2} \Phi(ax+b)\,\mathrm dx$$ Here $\Phi(\cdot)$ is the cumulative probability distribution function of a standard normal random ...
0
votes
0answers
20 views

The integration of a Gaussian process.

Now I'm reading this post: Distribution of integral of a normally distributed random variable Suppose $r(t),t\in[0,T]$ is a Gaussian process.I want to show that $$\int_0^Tr(t)\,dt$$ has normal ...
0
votes
1answer
21 views

composition of probability distribution functions

Suppose we are given $X \sim \mathcal{N}(\mu,\Sigma)$. Then, we define the random variable $Y$ as follows: $Y_i = 1 + X_i $ if $X_i \ge 0$ $Y_i = \exp(X_i)$ if $X_i \lt 0$. How do I go about ...
0
votes
0answers
19 views

Product of normal densities in a Bayesian context

Two analysts, analyst A and analyst B, are interested in the probability distribution for a multivariate-normal vector $X$ with five dimensions. A estimates a density function $f_X(X=x)$ for $X$, ...
1
vote
2answers
66 views

Histogram of random numbers from normal distribution

If I generate, say, 10000 numbers from the normal distribution (in Matlab) and want to draw a histogram with 10 bins, it resembles the normal distribution pretty accurately. However, if I decide to ...
2
votes
0answers
25 views

Distribution of sample statistics taken from bivariate normal

$(X_{1},Y_{1}),\,...\,,(X_{n},Y_{n})' (n>2)$ are random samples taken from $N_{2}((\mu_{1},\,\mu_{2})',\,$$ \begin{pmatrix} \sigma^{2}_{1} & \rho\sigma_{1}\sigma_{2} \\ ...
3
votes
2answers
52 views

If $X$ is standard Normal then find $\lim_{x\to0}P(X>x+\frac{a}{x}|X>x)$

If $X$ is Standard Normal and $a>0$ is a constant then find $\lim_{x\to0}P\big(X>x+\dfrac{a}{x}\big|X>x\big)$. This is an exercise from a book whose name I cannot immediately recall. I ...
2
votes
0answers
29 views

Distribution of some linear combination of Normal RVs

I would like to ask for help concerning this question lifted from the book An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.). Let $X_1$ and $X_2$ be independent ...
0
votes
2answers
37 views

Finding standard deviation from a normal distribution

This is for a homework assignment, I'm really only confused about one step within the problem though. Question: Suppose $X \sim N(400, \sigma^2)$ where $\sigma$ is the standard deviation. If we have ...