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

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1answer
51 views

Conditional density, bivariate normal

Let $Z=X+Y$ where $X \sim N(\mu,\sigma^2)$ and $Y \sim N(0,1)$ are independent. What is the conditional density of X given Z, $f_{X|Z}(x|z)$? I already found that ...
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1answer
43 views

Verification of linear combinations of a normal distribution

A machined part consists of 5 independent components connected end-to-end. Two of these have lengths $N(37.0, 0.49)$, and three of these have lengths $N(24.0, 0.09)$. All measurements are in mm. What ...
2
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3answers
89 views

How many tickets can you sell for a plane?

I'm trying to learn about limit theorems, but I have no idea how to calculate the following question: Suppose that 15% of people don’t show up for a flight, and suppose that their decisions are ...
2
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1answer
153 views

Covariance between squared and exponential of Gaussian random variables

Assuming the random vector $[X \ \ Y]'$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]'$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X ...
2
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1answer
33 views

Joint density of normal random variables

Let $Z=X+Y$ where $X$~$N(\mu,\sigma^2)$ and $Y$~$N(0,1)$ are independents. Find the joint density of Z and X. This is the first time I see something like that, look what I did below: I know ...
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0answers
39 views

Hamming weight in multiple label

Assume you have a $N$ balls. You give each ball $T$ different labels randomly from $\{0,\dots, N-1\}$. So hamming weight of each of labelling varies from $0$ to $\lceil\log_2 N\rceil$. What is ...
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0answers
40 views

Find the distribution of $Z=\frac{X_1+X_2}{X_1X_2}$, where $X_1$, $X_2$ follow normal distribution

Lets assume $X_1$, $X_2$ follow normal distribution. I am looking for the distribution of: $$Z = \frac{(X_1+X_2)}{X_1*X_2} $$ This is what I have thought so far: The distribution of the ...
-1
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1answer
29 views

Probability/Statistics help? [closed]

In a population, height of females are normally distributed with mean $\mu_1 = 162\mathrm{cm}$ and standard deviation $\sigma_1=6\mathrm{cm}$. Heights of males are normally distributed with mean ...
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0answers
23 views

T-ratio. Estimation of standard error.

Let $ X = (X_1, ..., X_n)$ be a vector observation collected from Normal Distribution. We don't know neither variance of population nor expected value. We would like to estimate expected value for ...
2
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1answer
99 views

Standard distribution formulae trick

I am trying to understand the following question. The height of adult males is normally distributed with a mean of 172cm and standard deviation of 8cm. If 99% of adult males exceed a certain height, ...
2
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1answer
30 views

Integral estimate: upper bound normal distribution

I'm looking at the proof of Donsker's theorem and it is used that $\frac{1}{\delta}\int_{\frac{1}{\sqrt{\delta}}}^\infty{e^{-y^2/2}}dy\leq \int_{\frac{1}{\sqrt{\delta}}}^\infty{y^2e^{-y^2/2}}dy$ ...
2
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2answers
47 views

Proving that the maximum values of these different, Normal distribution, curves are different.

In a question, one variable X is Normally distributed with mean=100, variance=25 and Y is Normally distributed with mean=110, variance=36. The question asks to sketch the p.d.f of each on the same ...
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2answers
41 views

Sum and difference of three normally distributed variables

We are given three independent random variables $X, Y, Z$ with normal distribution $\mathcal{N}(1,2)$. Are $U=Z-Y+X$ and $V=X+Y$ independent? I thought I would compute the joint density $f_{UV}$ and ...
0
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0answers
29 views

Maximizing the weighted sum of two CDFs subject to a constraint on the expected value.

I encountered this problem in a proof and would like to have your help: Consider the maximization problem: \begin{eqnarray} \max_{x,y}b_x\Phi(x)+b_y\Phi(y),s.t\\ ...
2
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0answers
34 views

Distribution of $\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$ given $\langle x, y\rangle$

Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors, and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$. Can we write the ...
1
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1answer
74 views

grading on a curve using normal distribution

suppose we have a grade list: $ \text{grades}=\{2,3,5,7,8,10,9,9.75,8,0,11,10,10,3,5.25,13,14,20,18,9\}; $ which mean equals to 8.75 and Standard deviation is 5.06471. we want to improve the grade ...
0
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1answer
60 views

statistics and biased estimator of normal distributions

Let $X_1, X_2 , X_3 , X_4$ and be independent, identically distributed random variables from a population with mean $\mu = 10$ and variance $\sigma^2 = 10$ . Let $\bar Y = \frac{1}{4}(X_1 + X_2 + ...
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0answers
74 views

Normal Exponential Convolution proof

I am seeing a scientific paper where they explain background correction modelled as two random independent variables one with exponential distribution and the other with normal distribution. ...
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0answers
53 views

Are Gaussians a basis for the vector space of continuous functions?

How can I prove (or disprove) that the Gaussian function family: $f_{\mu,\sigma}(x)=e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$ Are a basis for $C(\mathbb{R})$ ?
2
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1answer
99 views

Normally distributed variable with normally distributed mean.

What the idea behind the prove of following statement? I am pretty sure the statement it is correct. If $X \sim N(\text{mean}_x, \text{var}_x)$ and $Y \sim N(\text{mean}_y+X, \text{var}_y)$, then $Y ...
2
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0answers
46 views

Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta ...
0
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1answer
48 views

Relationship between univariate normal distribution and multivariate normal distribution

Let $a_1, a_2, a_3$ is column vector and $H = [a_1 a_2 a_3]$. If $a_i$ have standard normal distribution, is this following statement true ? $$ vec(H) = [(a_1)^T (a_2)^T (a_3)^T]^T$$ have multivariate ...
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0answers
178 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...
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1answer
29 views

Pdf of a normal variable accepted with probability dependant on the normal variable

Assume $z$ is a standard normal variable. If $z<0$, then we accept it with probability 1. if $z\ge0$, we accept it with probability $e^{-mz}$, where $m>0$. I'm trying to figure out the pdf of ...
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1answer
400 views

Variance of a quadratic form

I am considering a variance of two forms: $ R(x) = (x-m)^\top A (x-m) + b^\top (x-m) + c $ $ R'(\Delta) = \Delta^\top A \Delta + b^\top \Delta + c $ where $x$ is a random variable of ...
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1answer
15 views

Proporties of linear-combinations of normally distributed variables.

At my school, to pass an exam, you'll have to score at least 230 points. The results are normally distributed with $\mu=200$ and $\sigma=20$. If I were to consider 10 students who attends the exam, ...
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0answers
21 views

Height of a point

I wonder if there is way to find out the height of a specific point x in a normal distribution whenever the standard deviation is not given.?
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0answers
36 views

Find the distribution of $Z = 1/X_1 + 1/X_2$

Find the distribution of $Z = 1/X_1 + 1/X_2$, where $X_1$ and $X_2$ follow normal distribution. I have $2$ variables with normal distribution, $X_1$ and $X_2$. How can I find the distribution of: ...
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0answers
17 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
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1answer
58 views

Translate exponential distribution into normal distribution

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
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2answers
37 views

Proving the $Pr(d>0|a+d=\pi)$ is increasing in $\pi$ when a and d are two independent normal distributions.

I was wondering if it is possible to prove the following (or show false otherwise). Given two independently distributed random variables $a\sim \mathcal{N}(\alpha,\sigma_\alpha^2)$ $d\sim ...
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0answers
25 views

Calculation of arrival time of messages from 1 source through 2 different routes

I need to simulate sending messages from $A$ to $B$ as follows: Each message is sent $N$ times from $A$ on the same time, passes through a certain route $R_n$ and arrives at $B$. Travel time of $R_n$ ...
2
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0answers
47 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
0
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1answer
40 views

Variable drawn from a normal distribution

What exactly is the meaning of a "variable drawn from a normal distribution"? I know what a normal distribution is, but my main exposure to "variables" is from calculus, so I have a hard time ...
2
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3answers
119 views

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian. Mathematically, if we know that $a_1X+b_1Y$ and $a_2X+b_2Y$ have Gaussian ...
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0answers
36 views

Mixture of binomial distributions

I have a population of agents with a single real-valued attribute $x$. Each of them performs $n$ Bernoulli trials with success probability $q(x)$ which depends on their attribute. In particular, $$ ...
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1answer
39 views

Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
1
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1answer
61 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
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0answers
43 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
2
votes
1answer
105 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the symmetric semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes ...
2
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1answer
47 views

Are $X$ and $Y$ necessarily normal if the the sum $Z=X+Y$ is normal?

Of course, asking the question the other way round is straightforward to answer as via the convolution we find that the sum of two normal distributed variables is again normal. But however, is it ...
1
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2answers
41 views

Normal Distribution of Sums

I have two normally distributed random variables $X$ and $Y$. Then I know that the sum $X-Y$ is also normally distributed (i). However, I want to show (preferably by a counter example) that the ...
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0answers
46 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...
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1answer
126 views

What is a decision threshold and how does it apply to a statistical power?

I'm pretty confused on what is actually going on in this section with hypothesis testing. As another note, the values below are computed using R. I have a homework problem that says: From the ...
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1answer
205 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
0
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1answer
49 views

chi square distribution probability

I am having a problem with this. Suppose a stock's returns are normally distributed with mean $m$ and variance $\alpha^2$ and we compute the sample variance from a sample of $41$ periods and find ...
4
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1answer
87 views

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
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0answers
43 views

Distribution of unknown covariance matrix, given variance of linear combination

Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$? Specifically ...
1
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1answer
17 views

When the domain of a continuous distribution exceeds feasible values, what should I do?

Now I need a (maybe approximated) model for this distribution: $$X=(x_1, x_2, …, x_n)$$ where $x_i$ is a real number between $0.0$ and $1.0$, and the sum of $x_i$ equals $1.0$. Now, I want to use ...
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0answers
62 views

MLE of variance for a spherical Gaussian

I am trying to implement the X-Means clustering algorithm. In it, the authors use the BIC to determine which model fits the data best. It is explained here: ...