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

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13 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. ...
-1
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0answers
20 views

integral involving pdf of normal distribution

I want to calculate following integral any solution (closed form, or in terms of numerical functions, even approximations will help)
0
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1answer
21 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
13 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
42 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 ...
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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 ...
0
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1answer
16 views

composition of probability distribution functions

Suppose we are given $X \sim \mathcal{N}(\mu,\Sigma)$. Then, we define the random variable $Y$ as follows: $Y_i = 1 + X_i $ if $X_i \ge 0$ $Y_i = \exp(X_i)$ if $X_i \lt 0$. How do I go about ...
0
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0answers
15 views

Product of normal densities in a Bayesian context

Two analysts, analyst A and analyst B, are interested in the probability distribution for a multivariate-normal vector $X$ with five dimensions. A estimates a density function $f_X(X=x)$ for $X$, ...
1
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2answers
31 views

Histogram of random numbers from normal distribution

If I generate, say, 10000 numbers from the normal distribution (in Matlab) and want to draw a histogram with 10 bins, it resembles the normal distribution pretty accurately. However, if I decide to ...
2
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0answers
22 views

Distribution of sample statistics taken from bivariate normal

$(X_{1},Y_{1}),\,...\,,(X_{n},Y_{n})' (n>2)$ are random samples taken from $N_{2}((\mu_{1},\,\mu_{2})',\,$$ \begin{pmatrix} \sigma^{2}_{1} & \rho\sigma_{1}\sigma_{2} \\ ...
4
votes
2answers
48 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
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0answers
23 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
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2answers
20 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
69 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
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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} & ...
1
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0answers
10 views

Normalizing constant for product of Gaussian densities - interpretation

The normalizing "constant" for the product of two multivariate Gaussian densities, with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is (the reciproke of) a ...
1
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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 ...
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1answer
43 views

Proving conditional distributions are normal [closed]

For the standard bivariate normal distribution, it is easy to show (by simple integration) that both marginal distributions are N(0, 1). Prove that both of the conditional distributions are also ...
4
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0answers
34 views
+50

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
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0answers
4 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 ...
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0answers
37 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
95 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 ...
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1answer
22 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 ...
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0answers
27 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 ...
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0answers
19 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
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1answer
14 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
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0answers
18 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
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0answers
79 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)= ...
1
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1answer
34 views

What distribution it is based on the histogram? [closed]

I generated this histogram in r and was trying to determine which distribution I should use, my guess is normal or Binormial. But I'm not sure, can anyone help please?
0
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1answer
38 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
47 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
12 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
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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
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0answers
19 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
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1answer
16 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
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0answers
17 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
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2answers
20 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
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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 ...
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0answers
29 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
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0answers
40 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 ...
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0answers
16 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 ...
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1answer
22 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 ...
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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
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1answer
21 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 \\ ...
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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
21 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 $\sim \text{N}(0,(t-s)^2)$ ?
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2answers
26 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
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1answer
23 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: ...
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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 ...