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

learn more… | top users | synonyms

0
votes
1answer
46 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 ...
0
votes
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 ...
0
votes
0answers
36 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
votes
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
votes
0answers
54 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
90 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
0
votes
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
70 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
votes
2answers
50 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
39 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
83 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
113 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
votes
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
votes
1answer
64 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
33 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
votes
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
votes
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
121 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
vote
0answers
34 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
72 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
votes
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 ...
0
votes
0answers
26 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
votes
1answer
21 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
101 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
87 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
50 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 ...
0
votes
2answers
58 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 ...
0
votes
1answer
40 views

Contaminated normals, Multivariate normal distributions and PCA

While studying the above mentioned topics, i got a little confused in reading two things. I have two questions. First, in: Scanned text 1 how exactly P[W <= w] unfolds as seen in the red ...
3
votes
1answer
57 views

Integral of 2D Gaussian over triangle

I am interested in computing (numerically) the integral of a 2D Gaussian (with arbitrary mean and covariance matrix) over a triangle in the plane. I would like the solution to work over a wide range ...
1
vote
0answers
16 views

estimate normal distribution parameters by $n$ largest samples

If I have the $n$ largest out of $m$ values of a sample from independent normal distributed random variables $\mathbb{X}_1,\dots,\mathbb{X}_m\sim\mathcal{N}(\mu,\sigma)$ with unknown parameters ...
1
vote
1answer
23 views

Can I conclude the following about bivariate normal RV?

If $(X,Y)$ is bivariate normal with mean $[0, 0]$ and variance-covariance matrix $ \left[ \begin{array}{ccc}1 & \rho \\ \rho & 1 \end{array} \right]$ and $Z=-X$ then is it true that $(Z,Y)$ ...
0
votes
1answer
56 views

Differentiating mahalanobis distance

I would like to differentiate the mahalanobis distance: $$D(\textbf{x}, \boldsymbol \mu, \Sigma) = (\textbf{x}-\boldsymbol \mu)^T\Sigma^{-1}(\textbf{x}-\boldsymbol \mu)$$ where $\textbf{x} = (x_1, ...
0
votes
1answer
42 views

Comparing 2 Gaussian Distribution

I have 2 different dataset of about 1000 points each. Actually, the 2 are not so different generally. I want to compare the 2 data but my statically knowledge is quite poor. My idea is to construct ...
3
votes
2answers
111 views

Taking a derivative with respect to a matrix

I'm studying about EM-algorithm and on one point in my reference the author is taking a derivative of a function with respect to a matrix. Could someone explain how does one take the derivative of a ...
2
votes
1answer
38 views

multi-variate normal distribution distance from vector sub-space

let $X\sim {\cal N}(\mu,C)$ be a random variable obeying multi-variate normal distribution in $\mathbb{R}^n$ and $U \subset \mathbb{R}^n$ be a vector space with $\dim(U)=n-1$. What is the probability ...
0
votes
1answer
37 views

Probability that a $n$-dimensional Gaussian falls into a half-space

For $a \in \mathbb{R}_{\ge 0}^d$ and $b \in \mathbb{R}_{\ge 0}$, we can define a half-space as the set of points $x \in \mathbb{R}^d$ such that $a \cdot x \le b$, namely, $$\mathcal{H}(a,b) = \{x \in ...
1
vote
0answers
22 views

What is the probability the maximum sample value comes from one of two random distributions?

Let $X_1$ and $X_2$ be randomly distributed variables with means $\mu_1$ and $\mu_2$ and standard deviations $\sigma_1$ and $\sigma_2$. Samples of size of $n_1$ and $n_2$ are drawn from each ...
1
vote
2answers
114 views

how to show $E[|X|]= \sigma$ where $X \sim N(0, \sigma^2)$

let $X \sim N(0, \sigma^2)$ I want to show $$E[|X|]= \sigma$$ thanks for help
1
vote
1answer
55 views

Construction of Gaussian Hilbert spaces

I am reading the very first chapter of "Gaussian Hilbert Spaces" by S. Janson. Definition: A Gaussian Hilbert space is a closed subspace of $L^2(\Omega, \mathcal{F}, P)$ consisting of centered ...
0
votes
3answers
42 views

integrating $A^2=\frac{1}{2\pi}\int^\infty_{-\infty}\int^\infty_{-\infty}e^{-\frac{y^2+z^2}{2}}dydz$

When proving that $$\int^{\infty}_{-\infty}\frac{1}{\sqrt{2\pi}\sigma}{e^{-\frac{1}{2}({\frac{x-\mu}{\sigma})}^2}}dx=1$$ and I faced a problem, ...
1
vote
0answers
39 views

conditional expectation of squared standard normal

Let $A,B$ independent standard normals. What is $E(A^2|A+B)$? Is the following ok? $A,B$ iid and hence $(A^2,A+B),(B^2,A+B)$ iid. Therefore we have $\int_M A^2 dP = \int_M B^2 dP$ for every ...
0
votes
1answer
188 views

Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: ...
1
vote
1answer
71 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
0
votes
1answer
45 views

Why is it so easy to marginalize a multivariate random distribution?

From wikipedia: To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to ...
0
votes
0answers
59 views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, ...