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

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1answer
54 views

How can we derive expectation of two dependent normal distribution?

$\mathbf{X}$ and $\mathbf{Y}$ are each dependent normal random variable, then how can we derive like this one? $$\mathbf{E}\{e^{\mathbf{X}}e^{\mathbf{Y}}\}$$ I know the each first moment is ...
0
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1answer
21 views

Verify that moments of gaussian variable are given by a formula

I would like to ask you to verify if the following statement is true. Let $X$ be a normal-distributed R.V. with $0$ mean and $\sigma ^2$ variance. Then $$ \mathrm{E}\left[X^p\right] = ...
0
votes
1answer
16 views

statistics - multivariate normal distn, variance and probability of event?

I have a multivariate Normal distribution defined by: μx = 360, μy = 280, μz = 180, σx = 40, σy = 34, σz = 48, and correlations of ρxy = −0.41, ρxz = −0.34, and ρyz = 0.47. I am required to find ...
1
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0answers
13 views

Linear Gaussian system, covariance of the normalisation constant

If we have the following multivariate Gaussian distributions: $$p(x) = N(x|\mu_x,\Sigma_x)$$ $$p(y|x) = N(y|Ax + b, \Sigma_y)$$ Now how can you deduce p(y) ? p(y) is called the normalisation ...
3
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2answers
80 views

If $X$ is normal, is $\exp(X)$ still normal? How to find its mean and variance?

$X$ is a random variable for normal distribution: $X\sim N(\mu, \sigma^2)$. What is the mean and variance of $\exp\{x\}$? My attempt: $$E[\exp\{x\}]=\exp \{E[x]\} \text{, by the invariance ...
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0answers
9 views

Two Way Random Effect Model. $\mathbb E(\alpha_i\bar \epsilon_{i..})$?

$\alpha_i$ is random effect of $i$th level of factor $A$ and $\alpha_i\sim NID(0,\sigma_{\alpha}^2)$ $\epsilon_{ijm}$ is the random error term of $i$th level of factor $A$ and $j$th level of factor ...
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0answers
21 views

intergral of the product of 2 multivariate Gaussian distribution

Suppose there are the following relationships between $x,y,w$, $$\begin{align}p(x,y) &= N(\mu_1, \Sigma_1)\\ p(x\mid w) &= N(\mu_2,\Sigma_2)\end{align},$$ is it possible to compute $p(y\mid ...
1
vote
1answer
38 views

Normal and standard distribution

There is some details i don't understand in my book, here goes; Let $X \sim N(\mu,\sigma^2)$ and $Z\sim N(0,1)$ we know that: $$F_X(x) = \int\limits_{-\infty}^{x} \frac{1}{\sigma ...
2
votes
1answer
27 views

Bound on the $Q$ function related to Chernoff bound

For the function $Q(x) := \mathbb{P}(Z>x)$ where $Z \sim \mathcal{N}(0,1)$ \begin{align} Q(x) = \int_{x}^\infty \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2} \right) \text{d}u, \end{align} for ...
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2answers
32 views

What is the probability that a Chi-square distribution lies within 2 standard deviation of its mean?

Here I have an 8 degrees of freedom Chi-square distribution function $f(x)$ So by definition, $E(X)=8, Var(X)=2*8=16$. (Please guide me if this is wrong. We just started this chapter and there's ...
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0answers
18 views

How to generate normally distributed random numbers? [duplicate]

Is there any function that can generate normally distributed random numbers?
0
votes
1answer
108 views

'normally distributed random numbers' vs 'uniformly distributed random number'?

what is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)
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0answers
193 views

Distribution of the sum of squared independent normal random variables.

The sum of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then ...
2
votes
1answer
139 views

Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
0
votes
1answer
18 views

If each $Xi$ is distributed $N(0, \sigma^2)$, what is the distribution of $\sum_{i=1}^k X^2$?

I tried it for k=1 with an integral, so: $P(X^2 <x) = \int_{-\sqrt{x}}^{\sqrt{x}} \frac{1}{\sqrt{2 \pi} \sigma} e^{\frac{t^2}{\sigma^2}} dt$, but this didn't work out. I suspect there is faster ...
0
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1answer
51 views

how to generate Normally distributed random number?

I am looking for a function that can generate Normally distributed random numbers. I came to know about bux-muller transform but I didn't understood it completely what it is doing. Thus it would be ...
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0answers
5 views

Generating distribution from clusters

I am working on image processing where I have 15 clusters corresponding to 3 dimensional points. These points are clustered according to the 15 fixed variables over a duration. (for example 10 ...
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0answers
23 views

$(X, Y)^T$ ~ multivariate normal does NOT implies that X | Y $\in$ [a,b] has normal distribution??

I just found a rather surprising fact about the multivariate normal distribution. Suppose $(X,Y)^T$ has bivariate normal distribution; $$ \begin{bmatrix} X\\ Y \end{bmatrix} \sim MVN_2 \Big( ...
0
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1answer
47 views

Weighted mean from a set of average and standard deviation pairs

I'm trying to replicate some math a professor did related to Twitter sentiment analysis. Basically, there is a sentiment dictionary, called ANEW, that contains a mean and standard deviation for 3 ...
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0answers
34 views

Analytical computation of one Gaussian mixture model from another

I'm wondering if there is a way to analytically compute the optimal GMM (for a specific number of gaussians) in the case of approximating another GMM. E.g., is there an optimal single gaussian that ...
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0answers
37 views

Finding the joint posterior distribution of AR(2) process

Suppose we have AR(2) process for $\{y_t, t=3,4,..\}$ and let $a_1,a_2,\sigma^2$ be the parameters of the time series. We assume that $y_1$ and $y_2$ er independent normally distributed with mean zero ...
0
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1answer
30 views

Finding the z value

So I am new to normal distribution, I need help on understanding on how to calculate it's z-value Question : What is the z value which has 87.49% of the area below it?
2
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0answers
56 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
0
votes
1answer
112 views

Q function and the Chernoff bound

How do we use the Chernoff bound to prove that $$ Q(x)\leq e^{-\frac{x^{2}}{2}} $$ where Q(x) is the probability that a standard normal random variable X takes a value greater than x
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0answers
17 views

Sampling distribution with large sample size

As the sample size $n$ of a sampling distribution of sample means increases, the distribution becomes more normal. But if $n$ were the same size as the (finite) population, the "sampling" distribution ...
0
votes
2answers
73 views

Adding two normal distribution

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. And Suppose that $Z \sim N(1, 2^2)$ and is independent of all $X_i$. Define $Z_i = Z + X_i$ for $i = 1, ...
0
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2answers
51 views

Central Limit Theorem Application on Multivariate Normal

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. What is the distribution of $\overline{X} = \frac{1}{3}(X_1+X_2+X_3)$? I don't quite understand how to ...
1
vote
1answer
42 views

How is the entropy of the multivariate normal distribution with mean 0 calculated?

Here is what I have so far: $$\begin{align} h(x) &= - \int \frac{1}{(2\pi)^{\frac{D}{2}}\det\Sigma^{\frac{1}{2}}} \exp(-\frac{1}{2} x^T\Sigma^{-1}x) \ln ...
1
vote
1answer
35 views

The radial part of a normal distribution

I am reading a paper that asks me to sample $s_i$ from a distribution like this: $s_i \sim (2\pi)^{-\frac{d}{2}}A^{-1}_{d-1}r^{d-1}e^{-\frac{r^2}{2}}$ "Here the normalization constant $A_{d−1}$ ...
2
votes
0answers
84 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
3
votes
1answer
125 views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
0
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2answers
52 views

MGF/ expectation Gaussian Random Variables

I am stuck with something that seems easy but i cannot recall how to figure it out? Let $G_1$ and $G_2$ be two standard gaussian random variables with mean $0$ and variance $1$. Then how to calculate ...
0
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1answer
66 views

Combining discrete and continuous random variables

Here is the question: $X$ has the distribution $\mathcal{N}(0,1)$ and $Y$ is such that $P(Y=1) = P(Y=-1) = \frac{1}{2}$ Suppose that $X$ and $Y$ are independent and that $Z = XY$. Are $Y$ and ...
0
votes
1answer
34 views

Normal Distribution, cant seem to reach the right answer

The scores on a statistics test are Normally distributed with parameters mean = 80 and standard deviation = 196. Find the probability that a randomly chosen score is no greater than 70 My attempt, ...
0
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2answers
59 views

Random variables in normal distribution

Suppose that $X_1$, $X_2$ are independent $\mathcal{N}(0,4)$ random variables. Compute $P\left(X_1^2<36.84-X_2^2\right)$. I have no idea how to start this. Do I have to do anything to the ...
0
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0answers
22 views

Density of transformation of normal distribution

A data set contains real values $\left\{v_1,v_2,\text{...},v_k\right\}$, $k<\infty$. $X_n\sim \mathcal{N}(\mu ,\sigma ),\ n=1,2,...,k$ $P$ is the (not necessarily unique) permutation that ...
1
vote
2answers
123 views

Distribution of angle of two dimensional normal vector

The original subject is: Suppose random variables $X$ and $Y$ are independent and both follow the Normal distribution $N(0,\sigma ^2)$. 1) Prove $U=X^2+Y^2$ and $V = \frac{X}{\sqrt{X^2+Y^2}}$ ...
1
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0answers
35 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
2
votes
2answers
75 views

Standard Deviation greater than mean implies no normal distribution?

I understand that the mean $\mu \pm \sigma$ gives a better approximation of the measurements. But how is it related to the normal distribution? Is it because since $\sigma > \mu$ so the normal ...
0
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0answers
9 views

Normal distribution light-lifetime question

I have this in the practice questions for my upcoming exam, and we have just learned about Normal distribution today. The lifetime of a street lightbulb is $X ~ N(1,000,40,000)$ hours. Find the ...
0
votes
1answer
20 views

Calculating distribution over increments of a range?

If I have 100 units at a cost of 1 dollar and a current value of 1.25 dollars, I currently have a 25% profit. If that current value begins to drop I would like to begin selling off units over a price ...
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0answers
27 views

Generating correlated random numbers from Normal Distributions

If I have a sequence taken from X~N (μ1 , σ1 ). Is it possible to generate a sequence of numbers drawn from Y~N (μ2 , σ2) such that X and Y have correlation ρ?
2
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1answer
23 views

probability of arbitrary distribution that value is $> \mu+/-3\sigma$

A normal distribution has the property: $P(X>\mu+/-3\sigma)=1-99.71$ What is the probability $P(X>\mu+/-3\sigma)$ for an arbitrary distribution? (Is it actually possible to have some general ...
2
votes
1answer
104 views

Brownian Bridge as a Gaussian Process

Let $B=\{B_t:t\geq 0\}$ be a standard Brownian motion. Define the Brownian brige $X=\{X_t:t\geq0\}$ as $$ X_t=B_t-tB_1\quad t\in[0,1] $$ Show that $X$ is (i) Gaussian and find its (ii) mean and (iii) ...
3
votes
1answer
90 views

Is a constant (deterministic random variable) Gaussian?

Consider a constant $c$. Is this constant a Gaussian random variable (i.e. is $c\sim\mathcal{N}(c,0)$)? I realize a constant is easily described as a discrete random variable, but I wish to use ...
3
votes
1answer
42 views

Truncated Mean Squared

Suppose that $X_{\sigma} \sim \mathcal{N}(\mu,\sigma^{2})$. I am interested in whether $f(\sigma)=\mathbb{E} (X_{\sigma}^2 1_{\{X_{\sigma}>0\}})$ is monotonic in $\sigma$ for all $\mu$. I ...
2
votes
1answer
52 views

$\mathbb P $- convergence implies $L^2$-convergence for gaussian sequences

Consider $(X_n)_{n \in \mathbb N}$ a sequence of gaussian random variables whose limit in probability exists and is given by $X$. I was interested in showing that in this particular case we have ...
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2answers
62 views

Can normal distribution stats be used on this data?

Background: I'm analyzing operating times for "gadget". At some moments the operation times are very high (emergency situation), so the data has a lot of outliers: I have eliminated outliers using ...
3
votes
1answer
45 views

Inequalities for the tail of the normal distribution (Halfin-Whitt paper)

I am reading the famous paper by Halfin and Whitt, [1]. I'd like to prove remark (1) on page 575. The authors state \begin{align} \frac{\beta \alpha}{(1-\alpha)} = \frac{\phi(\beta)}{\Phi(\beta)} ...
0
votes
1answer
54 views

Countermonotonicity and minimum linear correlation coefficient

In an example exercise they question whether it is possible to construct a bivariate distribution of $LN(0,1)$- and $LN(0,4)$-distributed random variables, where $LN(\mu,\sigma^2)$ is the log normal ...