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

learn more… | top users | synonyms

0
votes
1answer
50 views

integral with pdf of a gaussian

$$ I = \int_{0}^{\infty} x \phi(x) dx $$ where $\phi(x)$ is the pdf of a normal distribution. Here I read that: If $X = \mu + \sigma U$ with $U$ a std normal, $$ I = E[\mu + \sigma U; mu + \sigma ...
-5
votes
0answers
44 views

What's the value of Epsilon?

This is one solution, which I'm learning on. I don't know why ε is 0,02.
1
vote
1answer
11 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
1
vote
0answers
32 views

Is there exist a Joint Density of Geometric Brownian Motion and Stopping time

Consider a stock price $S(t)$ which follows Geometric Brownian Motion (GBM), $$dS_t = \mu S_t dt + \sigma S_t d W_t, S_0 = s_0$$ where $\mu$ is drift, and $\sigma >0$ is volatility, $W_t$ is ...
-1
votes
0answers
47 views

Help!! Probability Question. SOA Exam P. [on hold]

I have a question from the practice test... In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference the true ages and the rounded age is assumed ...
0
votes
1answer
26 views

Normal distribution Z score

Problem: The observed error "E" in a series of measurements is normally distributed with mean of 0. Approximately 2% of error are -10 or less. Approximately what fraction of the measurements have ...
2
votes
1answer
27 views

Show that $d^T Z\sim N(d^T\mu, d^TVd)$ [duplicate]

Consider $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$ with $\mu=(\mu_1,\ldots,\mu_n)^T$ and $V=\text{Cov}(Z)$. Show that for $d\in\mathbb{R}^n$ it is $$ d^TZ\sim N(d^T\mu,d^TVd). $$ For me it ...
2
votes
1answer
22 views

Gaussian prior favors values closest to zero?

I am reading an article on Bayesian Logistic Regression, where they're using Logistic Regression, imposing a Gaussian prior (with mean = 0) on its parameters. They state that a Gaussian prior favors ...
0
votes
1answer
58 views

Given $f_X$. Integrate $\int_0^\infty \log_2 (x+1) f_X \, dx$.

Say $Y=Log_2[1+x]=g(X)$ and $f_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ ...
1
vote
1answer
44 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
1
vote
1answer
21 views

Show $\lim_{n\to\infty}\sqrt{n}\bigg(\frac{\sum_{j=1}^{n}X_j}{\sum_{j=1}^{n}X_j^2}\bigg)=Z$

Let $(X_j)_{j\ge 1}$ be independent, double exponential with parameter $1$. Show that; $$\displaystyle\lim_{n\to\infty}\sqrt{n}\bigg(\frac{\sum_{j=1}^{n}X_j}{\sum_{j=1}^{n}X_j^2}\bigg)=Z$$ where ...
0
votes
0answers
13 views

K Weighted Nearest Neighbour - Comparing Gaussians

This problem relates to a WiFi Indoor Positioning method - http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3812643/ The problem consists of the following steps: 1) In a database, we will have stored the ...
1
vote
0answers
16 views

Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
0
votes
1answer
25 views

How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$?

I was working on a research project that involves taking the integral of $$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} ...
1
vote
1answer
50 views

Determining whether random variables are independent

If I have two random variables as follows: 1) A Gaussian distribution of wifi signal strengths at a known point 2) A Gaussian distribution of wifi signal strengths at an unknown point (Note that ...
1
vote
1answer
30 views

Given a histogram, programatically, how do I find the normal distributions that comprise it?

I will be getting data in at around 100 frames per second, and I need to compute the normal distributions that comprise a set of 48 data points. The distributions can partially overlap, but will ...
1
vote
1answer
21 views

Normal Distribution and Probability on Excel

The size of fish in a lake follows a Normal Distribution with mean m = 1 lb 4 oz and standard deviation s = 3 oz . Fish that weigh less than 1 lb 9 oz must be released back into the lake. Bill ...
1
vote
2answers
65 views

Show that, $Z$ is $\mathcal N(0,1)$

If $Y\sim\mathcal N(0,1)$ and let $a>0$. Let $$Z=\ \begin{cases} Y&\text{if } |Y|\le a\\ -Y &\text{if }|Y|> a\\ \end{cases}\ $$ Show that $Z\sim\mathcal N(0,1)$ ...
0
votes
2answers
22 views

Calculating Variance

Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a population whose distribution is normal with mean $\mu$ and variance $\sigma^2$. Consider the statistics $\displaystyle T_1 = \frac{X_1 − X_2 ...
0
votes
2answers
41 views

Central Limit Theorem not valid?

According to Central Limit Theorem (CLT), the mean of any i.i.d. sample is Normal distributed (taking $n\rightarrow\infty$ samples). Let $X_i\sim U(a,b)$. Then $\bar{X}\sim N$ by CLT. But as we ...
1
vote
0answers
28 views

How to form Joint Probability Density from two Gaussian Distributions?

I've been reading the following paper entitled "An Improved Algorithm to Generate a Wi-Fi Fingerprint Database for Indoor Positioning": http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3812643/ In Part ...
0
votes
1answer
11 views

Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
0
votes
0answers
20 views

Gaussian MAX/MIN comparison

I've been reading the following paper entitled "An Improved Algorithm to Generate a Wi-Fi Fingerprint Database for Indoor Positioning": ...
1
vote
1answer
22 views

Expected value of normal distributed variable

I need to calculate the expected value of a modified normal distributed variable but i'm struggling. So maybe someone can help me. Suppose we've got a normal distributed variable $X \sim ...
0
votes
1answer
16 views

How to determine the distribution of $U:=(X,Y,Z)$?

I've got a question concerning the distribution of a multi dimensional random variable. I know that $X$ and $Y$ and $Z$ are each normal distributed with certain expectations and variances. ...
0
votes
1answer
29 views

How to determine the multivariate distribution?

Consider $$ Z_1:=\bar{Y}_1-\bar{Y}_2\sim N(0,\sigma^2(n_1^{-1}+n_2^{-1})),\\ Z_2:=\bar{Y}_1-\bar{Y}_3\sim N(0,\sigma^2(n_1^{-1}+n_3^{-1})),\\ Z_3:=\bar{Y}_2-\bar{Y}_3\sim ...
1
vote
1answer
34 views

Show that $E(S)=\sqrt{\frac{1}{n-1}}\frac{\Gamma(n/2)}{\Gamma[(n-1)/2]}\sigma$

Let $X_1,...,X_n$ be a random sample of size $n$ from the normal distribution with mean $\mu$ and variance $\sigma^2$ and let $S^2=\frac{1}{n-1}\sum^n_{i=1}(X_i-\bar{X})^2$ be the sample variance. ...
1
vote
1answer
54 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
2
votes
2answers
32 views

Distribution of a sum of normal distributions?

$X$ = weight of a small bag of crisps has Normal distribution with mean = $35.5$ and $var = 0.8$ . $Y$ = weight of a large bag of crisps has Normal distribution with mean = $152$ and $var = 3.2$ ...
0
votes
1answer
21 views

Normal Random Variables

Let Z1 and Z2 be independent standard normal random variables. What is the probability that the minimum of Z1 and Z2 will be greater than 1.0? How do I go about this when I have no values? Is the ...
-2
votes
1answer
23 views

Normal Random Distributions

A university expects to receive, for the next academic year, 16000 applications for admission to the bachelor’s degree program. The SAT score obtained by the applicants is modeled as a normal random ...
1
vote
1answer
42 views

Combining statistical distributions

I have a situation where a distribution is dependent on 2 variables, one of which follows the poisson distribution, and the other the normal distribution, and I want to establish the method of ...
-4
votes
1answer
38 views

If a random variable X has mean of μ and standard deviation σ…

, then what will be the mean and standard deviation of (X − μ)/σ ?
1
vote
0answers
27 views

Can this be solved analytically?

I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is $f(x) = 2 C_1 (X_1 - x) ...
4
votes
1answer
136 views

Why does this determinant have a continuous density at zero?

This question is a simplification of my previous question. I think this is easy, but I don't have a strong enough background in probability. Let $A$ be a random $n\times n$ real matrix that satisfies ...
2
votes
1answer
27 views

$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$ independent and normal

I am looking for a nice proof of the following statement: If $X,Y\sim U(0,1)$ are two independent uniformly distributed random variables, then $$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), \quad ...
0
votes
0answers
21 views

GEV of Normal Distribution and relationship of the parameters

My question goes on Extreme Value Theory for the Normal distribution: 1) Which GEV (Generalized Extreme Value distribution) type is the Normal distribution(Weibull/Gumbel/Frechet)? 2) If we have the ...
0
votes
1answer
26 views

How to add standard deviation regarding MATLAB function normrnd(mu,sigma)?

My question depend on this scenario which is as follows, I used a MATLAB function "normrnd(mu,sigma)" with mean 'mu = 0' and S.D 'sigma = 5', to generate a normal random number "R1". I added this ...
0
votes
1answer
44 views

Geometric BM tends to zero but is strictly positive a.s.?

The process $\{S_t\}_{t\ge0}$ following $dS_t = \sigma S_tdW_t$ with $S_0>0$ has the solution $$S_t=S_0 e^{-\frac12\sigma^2t+\sigma W_t}$$ Now for any $\epsilon>0$ we have $$\mathbb ...
2
votes
1answer
50 views

if $X_i$ are iid standard normal distributed, what is the limiting distribution of $\sum X^4 / (\sum X^2)^2$?

If $X_i$, $i=1,\ldots,n$ are iid standard normal distributed, what is the limiting distribution of $S_n=\sum X^4 / (\sum X^2)^2$? After finding the moments and since $Cov(X^4, X^2)=0$, I have the ...
0
votes
0answers
36 views

Problem on Expectation

Let $\Phi$ denote the standard normal distribution. Suppose $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. Show that ...
0
votes
1answer
36 views

Approximation in Normal distribution random variable

Let ${X_n : n \geq 1}$ be independent $\mathcal{N}(0,1)$ random variables. How do we get the following approximation?
0
votes
2answers
31 views

Given N-distribution, calculate expected value and Var of a function

$X_{{i}}=X_{{1}}...X_{{n}}$ is an iid. random variable with the distribution $N \left( {\frac {\alpha}{\beta}},{\frac {{\alpha}^{4}}{{\beta}^{4}}} \right) $ I would like to calculate the expectation ...
2
votes
0answers
19 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
0
votes
1answer
40 views

Find the probability that the average of X and Z is greater than Y. Where X, Z, and Y are normal RVs.

Here is the exact statement: Suppose X,Y , and Z are independent random variables. X is a normal random variable with mean 5 and variance 16, Y is a normal random variable with mean 7 and variance ...
0
votes
0answers
7 views

Moment generting function-Correlated Normal Variables

I have a variable x which is normally distributed with mean u1 and standard deviation sigma 1. We have another variable Y which is normaly distributed with mean u2 and standard deviation sigma 2. The ...
1
vote
1answer
33 views

Cumulative distribution function of a degenerate multivariate normal distribution

Let $X\in\mathbb{R}^{n}$ be a multivariate normal variable with the mean vector $\mu$ and the covariance matrix $\Sigma$. It is well known that if the matrix $\Sigma$ is positive-definite the ...
1
vote
2answers
26 views

MLE of MVN($\mu, \Sigma$)

I'm trying to find MLE of MVN($\mu, \Sigma$), i.e $N_k(\mu, \Sigma)$ with random sample $X_i, 1\le i \le n$. It was easy to get $\widehat{\mu}= \bar{X}$ and $\hat{\Sigma} = \frac{1}{n} \sum_i (X_i - ...
0
votes
2answers
29 views

Sum of maximum of two correlated normal random sequences

Let $x_{1},x_{2},\cdots,x_{n}$ and $y_{1},y_{2},\cdots,y_{n}$ be correlated normal random variables the covariance between two arbitrary random variables is known. In other words, let ...
1
vote
2answers
47 views

Concept of Probability in math first level

I am trying to teach myself the concepts of probability and I was wondering if this is correct. I am only 13 years old and did not learn this yet. I am just reading parts of a probability book to get ...