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

learn more… | top users | synonyms

1
vote
0answers
31 views

How to obtain the pdf of this transformation of normal random variables?

Given the following: Let $r = ab + n$, where $a, b,$ and $n$ are independent zero-mean Gaussian random variables with variances $\sigma_a$, $\sigma_b$, and $\sigma_n$, respectively. Find the MAP ...
1
vote
0answers
19 views

Bivariate normal exercise - check my answer please

Similar to the question I asked before, with one subtle difference. If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are correlated with $\rho = 0.5$ then find: $a)$ the covariance ...
1
vote
1answer
22 views

Bivariate normal exercise - check please

I am trying to self learn some probability and wanted to ensure I was getting these exercises correct. If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are independent, then find: $a)$ ...
1
vote
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 ...
1
vote
2answers
68 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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. ...
1
vote
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 $\mu_n$ ...
1
vote
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}) ...
1
vote
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} ...
1
vote
0answers
56 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{ ...
1
vote
0answers
32 views

population of dots with normal distribution of pitch

I want to generate a plot that shows a rectangle populated with dots, where the dot-to-dot distance (pitch) distribution is a lognormal (or a gaussian). I want to be able to change the mean dot-to-dot ...
1
vote
0answers
7 views

Conditional Covariance of a Normal conditionally autoregressive (CAR) prior

Suppose $$\bf{\nu}|\mu,\rho,\delta^2 \sim N_p(\mu \bf{1},\delta^2(D_w-\rho W)^{-1}),$$ where $W$ is a binary symmetric matrix and $D_w$ is diagonal with $(D_w)_{ii}=\sum_j w_{ij}$, $\mu$ is a scalar. ...
1
vote
1answer
54 views

Is this function increasing? (standard normal distribution, Mills Ratio)

Where $\phi\left(z\right)$ and $\Phi\left(z\right)$ represent the standard normal pdf and cdf respectively. 1) Is the function $f\left(z\right)=\dfrac{\phi\left(z\right)}{1-\Phi\left(z\right)}$ ...
1
vote
0answers
35 views

Show Almost Certain Convergence of a Sequence of Normal Random Variables

Let $(X_n)_{n=1}^\infty$ be independent, $N(0,1)$-distributed random variables. Prove that $$ \limsup_{n \to\infty}{X_n \over \sqrt{2 \log(n)}} = 1 \ \text{almost surely}.$$ I am aware of the ...
1
vote
1answer
93 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
1
vote
1answer
28 views

Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
1
vote
1answer
22 views

Is there a rule that can be used to easily approximate the pdf(x) for normal distribution?

Given the Normal Distribution with mean Mu and variance Sigma. With the respect to the rule of 3 Sigma, can one use similar estimations for the value of probability density function within 1, 2, ... ...
1
vote
0answers
22 views

How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$

Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard ...
1
vote
0answers
26 views

how to prove $\mathop {\lim }\limits_{n \to \infty } {\{\Phi [(1 - \varepsilon )\sqrt {2\log n} ]\}^n}=0$?

$\Phi (x)$ is the distribution function of standard normal distribution. $\varepsilon$ is some positive tiny number that is less than 1. How to prove this beautiful and important limitation: ...
1
vote
0answers
20 views

Explanation of Approximation for Integral Over Gaussian Distribution

I am reading an optics textbook that uses the following integral to evaluate the power squared in the lower tail of the following Gaussian integral. $$\frac{1}{{{\sigma _P} \cdot \sqrt {2 \cdot \pi } ...
1
vote
2answers
86 views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
1
vote
0answers
26 views

Empirical Rule. Is it applicable in this case?

So I ran in this problem: I have to test whether Empirical Rule is applicable. Proportions I got is 73%, 94,7% and 99.1% (within one, two and three standard deviations). I'm worried about 73%. This is ...
1
vote
0answers
29 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
1
vote
1answer
299 views

create a Gaussian distribution with a customize covariance in Matlab

the Matlab function 'randn' randomize a Gaussian distribution with $\mu= \begin {pmatrix} 0\\0\end{pmatrix}$ and $cov= \begin {pmatrix} 1&0\\0&1\end{pmatrix}$ Ineed to randomize a Gaussian ...
1
vote
1answer
22 views

Distribution of sample variance from normal distribution

Assuming $N$ samples $\{x_1,...,x_N\}$ are taken from a normal distribution with mean $\mu$ and variance $\sigma^2$, then the variance can be estimated using \begin{equation} ...
1
vote
0answers
22 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
1
vote
0answers
36 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $$\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$$ where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
1
vote
0answers
21 views

How to check $H_0$ hypothesis using Pearson's criteria?

How to check hypothesis by using Pearson's criteria ( $\chi^2$ test), that $H_0:$ random variable $X$ is normally distributed given that $k=7$ (count of intervals) and $\alpha=0.1 $ (significance ...
1
vote
0answers
33 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
1
vote
0answers
30 views

What is $E[\cos X]$ where $X$ is lognormal?

I was asked in an interview to compute $E[\cos X]$ where $X$ is lognormal. I tried using lognormal's characteristic function (Taylor series representation, which is divergent) and $\cos ...
1
vote
1answer
29 views

Applying a Normal Distribution to Another Function to Find Probability

Suppose that the number of hours students spend studying for an exam is approximately normally distributed with $\mu=10$ and $\sigma=\sqrt{2}$. If a student spends $t$ hours studying, he/she ...
1
vote
1answer
61 views

Box-Muller Transformation

I know that we can use the Box-Muller transformation to generate a pair of independent standard Gaussian random variables using a pair of independent standard uniform random variables. I am wondering ...
1
vote
0answers
24 views

Sampling Distribution of the Mean

I want to know if my reasoning is correct. Let's say I got two normal distributed variables: Variable "X": 5.4 (mean), 2.856 (variance) Variable "Y": 5.4 (mean), 5.062 (variance) Let's pick 16 ...
1
vote
3answers
115 views

Probability of two normal random variables when random samples are taken from a population

This is sort of second section to my previous question, I should have included both together, but I forgot to. Sorry for any inconvenience. X= random height of a male Y= random height of a female X ...
1
vote
0answers
52 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 ...
1
vote
1answer
38 views

Find percentage of conforming items for normal and uniform distributions

I'm given the following problem: Could anyone give some insight into how to solve this? I understand that for the normal distribution, I will likely have to look the percentage up in a table. ...
1
vote
1answer
22 views

Probability that two points (any where on the curve) are a set number of standard deviations apart on a normal distribution

So, here is the question: You buy two pieces of pipe from supplier A, and the inner diameter has a normal distribution of N(muA, sigmaA^2) = N(8.02, 0.1^2). You want these two pipes to butt together ...
1
vote
1answer
33 views

Normal distribution of independent and identically distributed variables

Suppose $X_1,...,X_n$ are independent and identically distributed $N(\mu,\sigma^2)$ random quantities. using the properties of independent normals and expectation and variance operators, explain why ...
1
vote
1answer
34 views

Calculate the asymptotic dystribution

Let $X_1,...,X_n$ be an i.i.d. random sample from a continuous distribution with density given by: $f(x;\theta)=(\theta-x)\frac{2}{\theta^2}$ if $0<=x<=\theta$ and 0 otherwise. We have the ...
1
vote
0answers
24 views

How to find value from Gaussian distribution for given point, covariance matrix and expected value.

While reading one article I came across that one of the values (probability) I am supposed to calculate is equal to N(v, b + (h^T)(W^T), I). Where b,v,h are vectors, W is a matrix and I is the ...
1
vote
0answers
28 views

Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
1
vote
1answer
41 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
1
vote
0answers
27 views

Range of sum of Normal Distribution.

May be its silly question but I was just wondering is there any way to find out the absolute range of sum of values of Random normal distribution of N numbers with mu and sigma as mean and Std. Dev. ? ...
1
vote
1answer
16 views

Normal distribution of juice

Quantity of juice in a pack of 1L is normaly distributed with average (mean) 950ml, and with standard deviation of 10ml. What is the probability that random pack of juice contains less then 945ml of ...
1
vote
0answers
21 views

What type of distribution can be used to describe a game with positive expected winnings?

I've come across something I'm not too sure about. Let's say we flip a coin, heads mean we lose 1 unit, tails means we win a 1 unit. This distribution of outcomes in this would be considered normal, ...
1
vote
0answers
25 views

Log-likelihood of the normal distribution.

On the attached picture I've highlighted the term which I do not agree with. Is it actually true ? In my calculations I get $$-n(\frac{1}{2}\log(\sqrt{2\pi})+\log(\sigma)),$$ instead. Thank you in ...
1
vote
0answers
55 views

Poisson process. Finding 5th and 95th centiles

I am an undergraduate student of Economics. Today I was trying to solve 1 exercise related to Poisson process that I found confusing and I would be very grateful for your help, as my Mathematics ...
1
vote
0answers
68 views

Ito formula for jump proccess

I have just learned Ito fomula for jump processes but I have still not understood it well. Assume that I have $dS_r=S_{t^-}\mu+S_{t^-}\sigma dB_t +S_{t^-}\int_{\mathbb{R}^+}(y-1)N(dt,dy), \;\; 0\leq ...