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

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4
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2answers
49 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
26 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
26 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 ...
0
votes
3answers
77 views

Finding a constant $ z $ such that $ P(Z \leq z) = 0.95 $ when $ Z \sim \text{N}(0,1) $.

This is for a homework assignment on normal distributions. Question: a) Find a constant $z$ such that $P(Z \leq z) = 0.95$ b) Find a constant $z$ such that $P(Z \geq z) = 0.95$ I'm having trouble ...
0
votes
0answers
12 views

Distribution between candidates

Lets say I have 5 candidates: 100 dollares has to be shared among them. Candidate 3 gets a least 35% of the money. Candidates: $$ \begin{array}{c|lcr} Canidate & \text{1} & \text{2} & ...
6
votes
0answers
86 views

Normalizing factor for product of Gaussian densities - interpretation with Bayes theorem

The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...
1
vote
1answer
60 views

Finding expected value of $|X_1+X_2|$, given that the two roots $X_1, X_2$ of $X^2 + 2BX + 1 =0 $ are real

$X^2 + 2BX + 1 =0 $ The random variable B is normally distrubuted with mean zero and unit variance. Given that the two roots $X_1, X_2$ are real, find, giving your answers to three s.f. the ...
9
votes
1answer
118 views

Characterization of normal distribution

I am sorry if this question is vague since I am completely unfamiliar with probability theory. Suppose that we have a family of real-valued random variables $X_n$ (say, all of them have mean 0) on ...
0
votes
0answers
5 views

Difference of two measurements (=means) from two normal distributions

I need help to understand which statistical test can be applied to test whether two subsequent measurements (from two different instruments measuring the same quantity) are signifcantly different from ...
1
vote
0answers
38 views

Exponent - Solving for an unknown within an expectation

I have reached a stage where I need to solve for an unknown number, $\theta$ . However, I stuck and don't know how to proceed further. The equation to be solved is: $E\left[ \exp(\theta a^i) * ...
3
votes
0answers
114 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
0
votes
1answer
24 views

“Distance” of iid gaussian variables [duplicate]

Take two i.i.d. Gaussian R.V.s $X$ are $Y$ both of which are $~N(0,a\sigma)$. Define a new R.V. $D = \sqrt{X^2 + Y^2}$. What's the expected value $E(D)$? In researching this I'm seeing references ...
0
votes
0answers
28 views

Bayesian Estimation of the mean of a multi-variate Gaussian

The posterior mean of a multivariate normal distribution is to be estimated with the Bayes rule for densities (http://www.math.uah.edu/stat/dist/Conditional.html), following the approach as described ...
0
votes
0answers
41 views

Normal Distribution finding q Question

The random variable X is normally distributed with mean 82 and standard deviation 7.4. Find the value of q such that P (82−q < X < 82+q) = 0.44. My friend sent me this answer. My doubt ...
0
votes
1answer
18 views

Variance of not quite the product of two independent, normally distributed random variables

Let's say I have two independent variables, $X\sim N(10,9)$ and $Y\sim N(5,4)$. $X$ represents the number of orders received in a month, and $Y$ represents the size of each order. For this example, a ...
1
vote
1answer
33 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
0
votes
0answers
83 views

Convert min to max probability

Assuming $Y=min(w_1,\ldots,w_n)$ , $w_i\sim N(\mu,\sigma^2) i.i.d$ I want to express $Y$ in terms of the $Q$ function. Knowing that $Y=min(w_1,\ldots,w_n)=-max(-w_1,\ldots,-w_n)$ $P(Y\leq y)= ...
0
votes
1answer
60 views

Product of two multivariate Gaussian pdfs - normalizing constant

https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf contains expressions (p.2, 6e, 6f) for the normalization constant for the product of two multivariate Gaussian pdfs, with mean vectors $a$ and $b$ ...
0
votes
1answer
52 views

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: ...
0
votes
0answers
13 views

What is the distribution of 'max of some normaldistributions'?

Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the ...
0
votes
1answer
25 views

How to approximate a normal distribution?

Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the ...
0
votes
0answers
8 views

Can I use Geometric Distribution to find the law of a total?

I have a variable X which is the amount of minerals in a dL(deciliter) of water. X follows a Normal Distribution X~N(μ,σ). I have the probabilty of the P(a ≤X< b) in a dl, where a and b are ...
0
votes
0answers
20 views

Is there any closed form for the integration of multiplication of two multivariate normal probability distributions?

I already computed the following integration but its a messy thing. I wonder if there is any easy way to compute it? or it has any closed form? V and p are known where V and p (p<1) are positive. ...
0
votes
1answer
18 views

Variance with minimal MSE in normal distribution

Given $X_1,...,X_n$ ~ i.i.d. $N(\mu, \sigma^2)$ where the mean is unknown, let the estimator for $\sigma^2$ be $\hat{e} = p\sum_{i=1}^n(X_i-\overline{X})^2$ How do I choose $p$ so that this estimator ...
0
votes
0answers
20 views

First order moment of multivariate Gaussian random vector

Let $X = (X_1,\dotsc, X_n)$ be a random vector distributed as a multivariate Gaussian with mean $0$ and covariance $\Sigma$. What is $\mathbb{E}[X_1\dots X_n]$?
0
votes
2answers
21 views

Covariance between $X$ and $Y$ of a bivariate normal distribution?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X$= 5 mL, $\sigma_Y$= 2 mL, $\mu_X$= 120 mL, $\mu_Y$= 100 mL, and $\rho$ = 0.6. How do I find the covariance of $X$ and $Y$? I know the ...
1
vote
0answers
24 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
1
vote
0answers
30 views

If B is a N(0,1) R.V., show $E[B^4] = 3$

I've read in Elementary Stochastic Processes by Mikosch (p. 98), that it is a well known fact that: If B is a N(0,1) R.V., $E[B^4] = 3$ I also see something equivalent (but uncited) on the ...
0
votes
0answers
54 views

Expected Value of the absolute value of the sum of random variables

Hi everyone and thanks in advance. Let's say we have a random variable Y which can be expressed as the sum of two other complex random variables X and W, i.e. $ Y = X + W $. $X$ and $W$ are ...
1
vote
0answers
17 views

Scaled distribution of Brownian motion

If I have $X = 5(B_t - B_s)$ Does this have a distribution of $\sim \text{N}(0,25(t-s))$ ? Since $B_t - B_s$ has distribution $\sim \text{N}(0,t-s)$ Then $X = \mu \cdot 0 + \sigma_1 Z$ where $Z ...
0
votes
1answer
24 views

Frequency Distribution and Throughput

I am conducting an experiment on a couple of computer systems but the results I have don't make sense to me. I made each system perform 1000 operations: System A performs operations at a rate of ...
0
votes
0answers
30 views

To determine asymptotic value of funtion for large N

To simplify the below expression as a function of m and N with m much less than N. $f(m,N)=\sum_{a=1}^{(N-m)/2} \binom{0.5*(N+m)-1}{a} \binom{0.5*(N-m)-1}{a-1} * (x)^{2a} * (y)^{N-2a} $ where ...
1
vote
1answer
23 views

How to extract a covariance matrix with this information

Referring to the above image, I wanted to know how to get the covariance matrix $\sum$. My understanding is, $A$, is our transformation matrice, such that $\begin{bmatrix} X_1 \\ X_2 \\ ...
0
votes
1answer
56 views

Product of two densities, when one of them is “incomplete”

One can frequently read that the product of two multivariate Gaussian pdfs, $f_1(x)$*$f_2(x)$, is itself a Gaussian function, with parameters as defined for example in: ...
2
votes
1answer
22 views

Linear transforms of Normal dist [closed]

If $X_t = \sqrt{t} Z$ where $Z \sim \text{N}(0,1)$ Then show the distribution of $X_t - X_s$ for $s<t$ Just wanted to check, would this be $\sim \text{N}(0,t-s)$ or $\sim \text{N}(0,(t-s)^2)$ ?
0
votes
2answers
29 views

Distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion

I am trying to find the distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ...
0
votes
1answer
26 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...
0
votes
1answer
14 views

Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
1
vote
1answer
25 views

distribution of distance between two points whose coordinates are normal random variables

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?
0
votes
1answer
38 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
0
votes
2answers
35 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
0
votes
1answer
57 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...
0
votes
1answer
100 views

inequality with gaussian cdf and density involved

in my calculations I've arrived at the following inequality $$ |\frac{4\phi(x)(1-2\Phi(x))}{(1+(1-2\Phi(x))^2)^2}| \leq 0.5 $$ where $\phi$ is Gaussian density, and $\Phi$ Gaussian cdf, which can ...
0
votes
1answer
55 views

Calculate P-Value

In a certain area, regulations require that the chlorine level in wastewater discharges be less than 100 $\mu$/L. In a sample of 85 wastewater specimens, the mean chlorine concentration was 98 ...
0
votes
2answers
50 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
0
votes
1answer
59 views

Expectation of a linear combinations of iid standard normal, restricted to a halfspace

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there any easy way to calculate $$\mathbb{E} \left[ 1_{\displaystyle \left\{ ...
0
votes
0answers
33 views

How to fit normal cumulative distribution functions

For a normal distribution $N(\mu,\sigma^2)$, we know its cumulative distribution function is $F(x)=\Phi(\frac{x-\mu}{\sigma})$ where $\Phi(x)$ is $cdf$ for standard normal distribution which means $$ ...
0
votes
1answer
42 views

Calculating probabilities for complex random variables

I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: ...
0
votes
0answers
42 views

The X & Y coordinates for points on a bell curve / normal distribution?

In Short: I want to give a formula the X coordinate and get the Y coordinate from matching a bell curve. Is this possible? In Detail: I'm trying to program a market simulation and to get a product's ...
2
votes
2answers
48 views

How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...