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
8 views

Variable drawn from a normal distribution

What exactly is the meaning of a "variable drawn from a normal distribution"? I know what a normal distribution is, but my main exposure to "variables" is from calculus, so I have a hard time ...
2
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3answers
23 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
16 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
11 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 ...
0
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0answers
15 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
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0answers
15 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 ...
1
vote
1answer
29 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the 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 $||X||$ ...
2
votes
1answer
40 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 ...
0
votes
2answers
32 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
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0answers
30 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
17 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
1answer
29 views

Is the product of $n$ Normal distributions also a normal distribution? [closed]

Is the product of $n$ Normal distributions also a normal distribution? I need not derivation just but answer.
0
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0answers
7 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 ...
-1
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0answers
31 views

What is your favorite approximation to the normal distribution? [closed]

I am asking this because my favorite is this one, which I independently discovered: $$N(x, 0, 1) =\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2} dt \approx \frac{1}{1+\exp(-ax)}, \text{ where } ...
0
votes
0answers
18 views

R Studio help - qqnorm [closed]

I'm loading in a file called data and using the command qqnorm like this data <- read.table("data.txt", header=TRUE) qqnorm(data) And get the error message "Error in xy.coords(x, y, xlabel, ...
3
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0answers
28 views
+50

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
38 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 ...
2
votes
0answers
36 views
+50

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
19 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
10 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 ...
-2
votes
0answers
21 views

The mean deviation from mean in a normal distribution is equal to $4\sigma/5$ [closed]

Show that the mean deviation from mean in a normal distribution is equal to $4\sigma/5$. Progress. I have tried going by the usual definitions of deviation and mean deviation but am stuck. Tried ...
1
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0answers
17 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: ...
0
votes
2answers
86 views

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

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 ...
0
votes
1answer
16 views

Sum of normal and log-normal independent random variables [closed]

X has a normal distribution, and Y has a log-normal distribution. X and Y are independent random variables. What is the distribution of X+Y? Thank you.
1
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2answers
17 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
vote
0answers
24 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
24 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
52 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
11 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
19 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
23 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
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1answer
26 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
35 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
40 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
11 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
51 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
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0answers
24 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
42 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
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0answers
16 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
15 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
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0answers
21 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
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2answers
29 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
51 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
26 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. ...
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votes
0answers
28 views

normal distribution, with mean of 58 and standard deviation of 9…

(a) The Statistics examination mark for students follows normal distribution, with mean of 58 and standard deviation of 9. (i) If the passing mark is 50, find the percentage of students who fail. ...
0
votes
1answer
22 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
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0answers
24 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
50 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
13 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 ...