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

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2
votes
2answers
126 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
1answer
26 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
1answer
53 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
31 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
397 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
192 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
52 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
85 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
103 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 ...
2
votes
1answer
60 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
94 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
69 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
139 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
21 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
27 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
167 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
48 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
443 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
43 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
57 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
28 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
154 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
141 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
49 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
53 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
469 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
197 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 ...
1
vote
1answer
89 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 ...
1
vote
2answers
277 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, ...
9
votes
1answer
283 views

length of Gaussian Random Vector

Suppose I have a random vector $x=[x_1,...,x_k]$ s.t. $x∼N(\mu,\sum)$. How is the length or magnitude of $x$ distributed? I know that if $k=2$ and $\sigma_1=\sigma_2$ and $\sigma_{12}=0$ ($x_1$ and ...
0
votes
2answers
55 views

What is the characteristic function used for?

Im totally new to statistics , but what is the characteristic function for ? I do not get that. I was reading about the bell curve and the Central Limit Theorem , but I did not get what the ...
0
votes
1answer
26 views

Stats probability

Is there a function in Excel that can calculate the following question? If not can someone tell me how it would be calculated? The answer should come out to 0.0228 "The average score of all pro ...
0
votes
1answer
201 views

Normal Distribution Stats percentage

I already know the answer of the following question, but not able to figure out how it's done. If someone can tell me it would be highly appreciated. "The owner of a fish market determined that the ...
1
vote
1answer
159 views

Example calculation of estimating GMM parameters using EM

I'm trying to study expectation maximization and I've almost got the idea. What I'm missing is a concrete example. Could someone familiar with the subject give me a concrete example how one would ...
1
vote
0answers
20 views

proof of As ~ N(A$\mu$, A$\Sigma$A')

assume that s is a vector of states which is distributed according to a gaussian with mean $\mu$ and variance $\Sigma$. A is the state transition matrix How can I proof that As ~ N(A$\mu$, ...
0
votes
1answer
165 views

How to calculate the a Probability with Z-Score.

I have a $ \mu = -1 $ and $ \theta = 6 $. I am supposed to find the probability $P(5 < X < 11)$. My Attempt: $$P(5< X < 11) $$ $$P(\frac{5--1}{6} < \frac{X--1}{6} < ...
1
vote
0answers
63 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
1
vote
1answer
122 views

Normally distributed from 0 to 99th percentile

Assume the length of waiting at supermarket is approximately normally distributed with mean 6 minutes and standard deviation 1.5 minutes. (1) What length of the waiting time constitutes the 99th ...
0
votes
2answers
57 views

How to get the lowest passing grade in normal distribution

$$ Pr[X\le a]=0.05 $$ $$ P[{Y\le {{a-64}\over 7.1}]} = 0.05 $$ I tried up to here but I don't know what to do now... Hint please.
1
vote
1answer
130 views

Showing that two Gaussian processes are independent

Say that $Z_t = (X_t, Y_t)$ is a 2-dimensional Gaussian process (by definition, it means that the random vector $(X_{t_1},Y_{t_1},...,X_{t_n},Y_{t_n})$ is a Gaussian random vector for all $t_1 ...
0
votes
1answer
98 views

Integral of Gaussian ring/shell

I would like to know the integral of the function $$f(\mathbf{x}) = {1 \over \sqrt{2\pi \sigma^2}} \exp\left\{- {(|\mathbf{x}| - \mu)^2 \over 2\cdot \sigma^2}\right\} $$ over an $n$-dimensional ...
1
vote
3answers
262 views

On Pr(X>Y) when X and Y are independent normal [duplicate]

Let X∼N(6,1) and Y∼N(7,1) be two independent normal variables. Find Pr(X>Y). the answer is 0.2389 but I do not know how to do it.I have tried adding them and subtracting but i am still clueless.
0
votes
1answer
35 views

Probability question of independent random varaibles

Let $X\sim \mathcal{N}(6,1)$ and $Y\sim\mathcal{N}(7,1)$ be two independent normal variables. Find $Pr(X>Y)$. the answer is $0.2389$ but I do not know how to do it.
0
votes
1answer
291 views

Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal ...
2
votes
1answer
220 views

Finding new probability density function with change of variable Y=sqrt(X)

Say we have a given distribution, such as X~No(a, b). I am trying to find the pdf and mean for $Y=\sqrt{X}$. I know the steps for finding the PDF, but since Y can only take on positive values, then ...
0
votes
1answer
46 views

Error function property

I have a question regarding a property of the error function. Is $k\cdot\text{erfc}(-x) = 1-k\cdot\text{erfc}(x)$ for all real $x$ for any $k$?
5
votes
1answer
361 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
1
vote
1answer
118 views

Sum of Gaussian Variables may not Gaussian

I am currently trying to understand the following three points which we discussed in lectures recently: We say that $X=(X_1,\ldots,X_d)$ is $d$-dimensional multivariate Gaussian distributed if ...
1
vote
2answers
103 views

How to use Chebyshev Inequality

Use Chebyshev Inequality to estimate the probability that in any one day of a business that earns a mean of 100 dollars a day with a standard deviation of 28.87 dollars, that business will make either ...
1
vote
1answer
55 views

Calculating the distribution of the average height - normal distribution

I am not sure how I am supposed to work this question out but I am given that the height of students from college A have a distribution written as: $A$~$N(1.78,0.06^2)$ and the height of students ...