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

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24 views

Is this Integral transformation correct?

I have an Integral: $$ \int_{-\infty}^{-y_1} \Phi(y_2)d\Phi(x_1) $$ Here: $\Phi(y_2)$ is the Gaussian density function of variable '$y_2$' which has to be integrated w.r.t Gaussian density of ...
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0answers
18 views

Proving a process is a P Brownian Motion

Let $X_t = tW_{\frac{1}{t}} \forall t>0$ and $X_0 = 0$. I am trying to show that this process is a brownian motion under some measure P. I have shown that it is continuous and that it is ...
5
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2answers
122 views

Convolution with Gaussian, without distribution theory, part 1

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
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0answers
4 views

Parameters of gaussian distribution, which is generated using central limit theorem

In a software I am working on (sensor simulation), I needed to generate normally distributed noise for simulated sensor signals. I used the central limit theorem. I generated 20 random numbers and ...
0
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0answers
30 views

Finding the expected value of a A Gaussian voltage distributions?

A Gaussian voltage random variable $X$ has a mean of $ \over X $ = 0, and variance of $9$. The voltage $X$ is applied to a square-law, full-wave diode detector with a transfer characteristic $Y = ...
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0answers
15 views

Bivariate normal distribution: showing that linear combinations of joint Gaussians are Gaussian

Refreshing my stats, I wanted to learn how to derive the bivariate normal distribution, for which I found this source on Wolfram.com. On that website, the authors derive the joint probability ...
4
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0answers
90 views

Joint distribution of the signs of the partial sums of independent standard normal random variables

Consider some i.i.d. standard normal random variables. What is the joint distribution of the signs of their partial sums? More formally, define a sequence of random variables ...
1
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2answers
67 views

Sum of a Normal and a Truncated Normal distribution

I have normal distribution $ N(\mu_1, \sigma_1)$ which shows the amount of demand in warehouse 1. The current amount of stock in the warehouse 1 is C. If the random demand is greater than C, it cannot ...
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0answers
24 views

Conditional PDF of multivariate normal distributions

Suppose that Y~$N\begin{pmatrix} 1\\ 2\end{pmatrix},\begin{pmatrix}2 & 1\\ 1 & 2 \end{pmatrix}$. How can I find the conditional PDF of $Y_1$ given that $Y_1+Y_2=3$?? I am given a hint to ...
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0answers
31 views

Proving chi-square distribution of estimator

There is something I can't get around my head. Let's suppose we havfe $ln x$ that is following a Normal distribution of parameter $lnx\rightarrow N(v;\theta )$ So we know that its estimator: $\hat ...
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1answer
24 views

Calculating standard deviation from a set of data

I'm trying to create a normal distribution of numbers between 0 and 100. I know that the mean = 28, and the only other information about the data is that there is a 10 % change that the number is 44, ...
0
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1answer
46 views

Why is Kurtosis of ND 3?

3 seems to be an important number when it comes to kurtosis. I see that it is often removed from the value entirely and this seems to be due to its being the kurtosis of the normal distribution. I ...
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3answers
34 views

Finding constant for CLT normal distribution

(This is from my textbook, but I don't understand their explanation. I've Googled around, but haven't found an answer that makes sense.) $$ \mu = 0, \sigma^2 = 1, n =16 $$ Find c such that: $$ ...
1
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2answers
40 views

Determine $A$ such that $Q=X'AX$ has chi-squared distribution.

Let $\boldsymbol X\sim N_n(\boldsymbol\mu,\boldsymbol\Sigma)$, where $\boldsymbol\Sigma$ positive-definite. I am trying to determine, in general, what form $\boldsymbol A$ (one example is ...
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1answer
24 views

Complex circular symmetric Gaussian and real Gaussian

Circular symmetric complex Gaussian zero mean PDF is defined as : $$f(z)= \frac{1}{\pi^N||M||} e^{-z^*M^{-1}z} $$ where $M$ is hermitian semi positive definite, $z \in \mathbb{C}^{N \times 1}$ and ...
1
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1answer
26 views

Square of Normal distributed variable?

This is a quick question. If $X\sim\operatorname{Normal}$, is $X^2$ rayleigh distributed? I ask this question is because from wiki, it says $X^2$ is called the Chi-Squared Distribution with a degree ...
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0answers
45 views

Random numbers generator

If I know how to generate random numbers from Gaussian distribution (using Box-Muller method), how can I generate random numbers from distribution with pdf ...
0
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1answer
15 views

Limiting Behaviour of Root Mean Square Normal Random Variables - Related to Chi-Squared Distribution

Above is my question. I have done the first part - made hard work of it, albeit, but still, it's done. The next part is where I am stuck. Intuitively, it seems (to me!) like we should have $R_n ...
-1
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3answers
27 views
0
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0answers
16 views

Non-linear transformation of symmetric distribution to get non-negative skewness

Say you have a variable $x \sim D(\mu,\sigma^2) $, where $D$ is a symmetric known distribution. I'm looking for two linear or non-linear transformations of $x$ that give one negative and one positive ...
0
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0answers
24 views

When computing normal distribution values in applications, should one round initial value?

I know how to compute the basic normal distribution problems but I have a question on rounding. The problem is We have bags of candies with average weight 150 g and standard deviation 5 g. The ...
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0answers
5 views

Conjugate Gaussian Prior

Suppose we have a univariate Gaussian distribution as our likelihood, and now we have our prior belief to be multivariate Gaussian distribution on our parameters, so what will the posterior be? Is it ...
2
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1answer
26 views

Proof of mean and vector

Let $X_1,\ldots,X_n$ be a random sample from $N(\mu, \sigma^2)$. Show that the sample mean and each $X_i-\bar X, i= 1,\ldots,n$, are iid. Actually $\bar X$ and the vector $(X_1-\bar X,\ldots,X_n-\bar ...
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0answers
16 views

Integrating to find area under probability density function of skewed distribution.

From flawr's answer to this earlier question of mine you can see that the probability density function of a skewed distribution is: $$f_a(x) = \frac{\phi(x)\Phi(ax)}{\Phi(0)} \qquad (*)$$ where ...
0
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1answer
19 views

Probability inference of an action from a continuous outcome

Assume person A takes an action, it could be either $a_1$ or $a_2$ with $a_1>a_2$, we cannot observe A's action but a signal $x$, with $x=a_i+\epsilon$. $\epsilon$ follows a normal distribution ...
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0answers
13 views

Multivariate Normal Density Concavity

For this variance compunent model $Y$~$N(X\beta, \Omega)$, where $\Omega=\sum_{i=1}^m\sigma_i^2V_i$, the log likelihood function is $(\beta, \sigma_1^1, ..., \sigma_m^2)=C+\frac12\log ...
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0answers
31 views

Convergence in distribution of normal random variables

Let $X_n \sim \mathcal{N}(\mu_n,\sigma_n^2)$. Prove that if $X_n \rightarrow X$ in distribution, then either $X$ is normally distributed or there exists a constant $c$ such that $X = c$ almost surely. ...
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1answer
25 views

How do you prove 2 normal random variables X and Y are jointly normally distributed?

How do you prove 2 normal random variables X and Y are jointly normally distributed? I know that any linear sum of X and Y should be normally distributed but how do you prove that?
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1answer
22 views

Sample Size Estimation with unknown variance

After spending a lot of time I'm still nowhere. What I have: Let d denote the desired margin of error so that the interval for µ is of width 2d. Let (1 − α) denote the probability associated with ...
0
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0answers
27 views

Worst case for $n$ Poisson trials?

I have $n$ Poisson events which occur with parameter $\lambda$. What can I expect the lowest of these to be? I'd be happy with any reasonable interpretation of the question, including "what is the ...
0
votes
1answer
48 views

Why erf(a-b)+erf(a)+erf(a+b) is so close to 3erf(a)?

I am approximating an empirical distribution function with a sum of three gaussians, and noticed that for erf function $erf(a-b)+erf(a)+erf(a+b)$ is numerically very close to $3erf(a)$ for applicable ...
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2answers
34 views

Convolutions and the Gaussian distribution

Suppose $X_1$ and $X_2$ are independent random variables each with the standard Gaussian distribution. Compute, using convolutions, the density of the distribution of $X_1 + X_2$ and show $X_1 + X_2 = ...
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0answers
10 views

The variance of a multivariate normal random variable

Suppose $\vec{X}$ is an N-dimentional random vector that is multivariate normal distributed: $\vec{X} = [X_1, X_2, ..., X_n]^T$ and $X_i \sim N(0,s_i^2)$ and all correlations bewteen $X_i$ and $X_j$ ...
0
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0answers
22 views

Moment List for Standard Normal Distribution

I am stuck trying to find the moment list for a standard normal. I have been told I can find it the similar way for exponential distributions using taylor series. I know the MGF = e^((1/2)(t^2)) for ...
1
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2answers
47 views

Find a 95% confidence interval on a binomial process.

Let's say that $73\%$ of $1506$ people interviewed were in favor of legalizing gay marriage. What is the $95\%$ confidence interval for the percentage of the public that are in favor of legalizing gay ...
0
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1answer
25 views

Normal distribution

X follows a regular normal distribution on V with center $\xi$ and inner product $<\cdot,\cdot>$, and let $\eta \neq 0$ be a vector in V. Show that the reel stochastic variable ...
0
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0answers
29 views

What is the probability of two things happening at the same time?

I am using the normal distribution for two events so there is a 34% chance of each event having one standard deviation above the mean. What is the probability of both events having one standard ...
1
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1answer
48 views

Limit of Gaussian random variables is Gaussian?

Consider a sequence $X_n$ of Gaussian random variables with mean $\mu_n$ and variance $\sigma_n^2$, which converges in distribution (to some limiting distribution). Can I then conclude that $\mu_n$ ...
0
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1answer
80 views

Finding Moment Generating Function of Normal Distribution

I need to show that the moment generating function of $Y$ is $$M(t)=(1 − 2σ^{2}t)^{−1/2}$$ where $X$ ∼ $N$($0$, $σ^{2}$) and that $Y$ = $X^{2}$. I know the moment generating function of $Z$ is ...
0
votes
1answer
12 views

Covariance in normal lognormal (NLN) mixture

Let $u = \epsilon e^{\frac{1}{2} \eta}$ where \begin{equation*} \left( \begin{array}{c} \epsilon \\ \eta \\ \end{array} \right) \sim N\left( \left( \begin{array}{c} 0 \\ 0 \\ ...
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2answers
24 views

Covariance of normally distributed random variables

If $ X \sim N(0,1) $ and given $ X = x $ then $ Y \sim N(x,1) $ I want to find the $ Cov(X,Y) $ using the relationship stated above. My attempt: $ Cov(X,Y) = E[XY] - E[X]E[Y] \\ E[X] = 0\\ ...
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0answers
32 views

Expected Z score for tail end of a standard normal distribution

Good day, For a Standard Normal Distribution (Mean: 0 and Variance: 1), I need to find the expected z-score for the section that makes up the final 2.5% (0.025) of the distribution. So, finding the ...
2
votes
1answer
28 views

Find marginal distribution of $Y$ where $Y\mid X$ is $N(a_1+a_2X,\sigma_1^2)$ and $X$ is $N(\mu,\sigma^2)$?

Let a random variable $X$ be normal $N(\mu,\sigma^2)$ and let the conditional distribution of $Y$ given $X$ be normal $N(a_1+a_2X,\sigma_1^2)$. a)Find the joint probability density function of $X$ ...
0
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1answer
25 views

Log-normal distribution

If $X \sim LogN(\mu,\sigma ^2) $, would the distribution for $aX \sim LogN(\mu+a,\sigma ^2) $ for $ a>0$? My solution: $log(X) \sim N(\mu,\sigma ^2) \\ log(aX) = log(a) + log(X) \\ log(aX) \sim ...
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0answers
26 views

A interesting question about moments.

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...
0
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1answer
46 views

Supremum of a sequence of i.i.d. standard normal random variables [closed]

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. standard normal distributed random variables. Show, using Borel-Cantelli's Lemma, that $$\sup_{n\in \mathbb{N}} |X_n|= \infty ...
0
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2answers
44 views

The probability involving the ordered normal sample.

Let $X_1,X_2,X_3,X_4,X_5$ be a normal sample, taken from the distribution with unknown mean $\mu$ and known variance $\sigma^2$. Calculate $$P(|X_{3:5}-\mu|<0.841\cdot\sigma).$$ Comment. If the ...
1
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0answers
26 views

How to find the compound of poisson and normal distribution?

how to find the compound distribution, if the rate of poisson distribution is normally distributed with mean and variance ? I know I have to find the integral of: $$ \frac {1} {\sigma \sqrt{2 \pi} ...
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0answers
54 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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0answers
19 views

Is a circular and spherical gaussian the same thing?

This pdf https://www.cs.princeton.edu/courses/archive/spring07/cos424/scribe_notes/0419.pdf mentions "circular gaussians" as the simplest gaussian in 2 dimensions, what you get if you have I as the ...