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

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33
votes
4answers
56k views

Is the product of two Gaussian random variables also a Gaussian?

Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed? Is the answer any different if we know that $X$ and $Y$ are independent?
43
votes
2answers
19k views

Expectation of the maximum of gaussian random variables

Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large? ...
13
votes
1answer
24k views

Calculation of the n-th central moment of the normal distribution $\mathcal{N}(\mu,\sigma^2)$

Since integration is not my strong suit I need some feedback on this, please: Let $Y$ be $\mathcal{N}(\mu,\sigma^2)$, the normal distrubution with parameters $\mu$ and $\sigma^2$. I know $\mu$ is the ...
7
votes
3answers
22k views

How to calculate the Fourier transform of a Gaussian function.

I would like to work out the Fourier transform for the Gaussian function: $f(x)=\exp(-n^2(x-m)^2)$. It seems likely that I will need to use differentiation and the shift rule at some point, but I can'...
3
votes
2answers
114 views

Correlated joint normal distribution: calculating a probability

Given $$ f_{XY}(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp \left( -\frac{x^2 +y^2 - 2\rho xy}{2(1-\rho^2)} \right) $$ $Y = Z\sqrt{1-\rho^2} + \rho X$ And $$ f_{XZ}(x,z) = \frac{1}{2\pi } \exp \...
1
vote
1answer
604 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
2
votes
5answers
3k views

The sum of $n$ independent normal random variables.

How can I prove that the sum of $X_1, X_2, \ldots,X_n$ random variables, all of which have normal distributions $N(\mu_i, \sigma_i)$, is a random variable that is itself normally distributed with mean ...
2
votes
2answers
1k views

Probability that the sum of all values of 5 pairs of dice will be between 30 and 40

I'm trying to solve a question that asks: If 5 pairs of fair dice are rolled, approximate the probability that the sum of the values obtained is between 30 and 40 inclusive. My approach so ...
35
votes
2answers
8k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...
8
votes
4answers
8k views

Derivation of the density function of student t-distribution from this big integral.

My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am ...
8
votes
3answers
369 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
8
votes
1answer
10k views

Probability of a point taken from a certain normal distribution will be greater than a point taken from another?

Let's say I have one point that will be taken randomly from a normal distribution with mean $\mu_1$ and standard deviation $\sigma_1$. Let's say I have another point that is taken much in the same ...
11
votes
1answer
7k views

Affine transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of $\mathbf{x}$....
3
votes
0answers
111 views

Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (actually, the mean ...
4
votes
1answer
2k views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
47
votes
9answers
9k views

What do $\pi$ and $e$ stand for in the normal distribution formula?

I'm a very beginner in mathematics and there is one thing I've been wondering recently. The formula for the normal distribution is: $$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\displaystyle{\frac{(x-\mu)^...
13
votes
3answers
33k views

How to calculate the integral in normal distribution?

The factory is making products with this normal distribution: $\mathcal{N}(0, 25)$. What should be the maximum error accepted with the probability of 0.90? [Result is 8.225 millimetre] How will I ...
16
votes
1answer
7k views

Characteristic function of a standard normal random variable

The characteristic function of a random variable X is given by $$\Phi_X(\omega) = \mathbb{E}e^{j\omega X}=\int_{-\infty}^\infty e^{j\omega x}f_X(x) dx.$$ One can easily capture the similarity between ...
15
votes
3answers
293 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\,\det\left(\pi^{-1}{A}\right)\right|^{-1/2}=\pi^{n/...
5
votes
2answers
2k views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, \...
2
votes
1answer
113 views

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose variance is distributed Log ...
2
votes
2answers
3k views

Expected value for maximum of n normal random variable

Let $X_1...X_n\sim N(\mu,\sigma)$ be normal random variables. Find the expected value of $\max_i(X_i)$ and $\min_i(X_i)$. The sad truth is I don't have any good idea how to start and I'll be glad ...
5
votes
1answer
6k views

Distribution of the sum of squared independent normal random variables.

The sum of squares of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then $X_1^2+...
8
votes
1answer
218 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 $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right|$$...
2
votes
1answer
195 views

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution

I am trying to derive Chi-square distribution. The random variale is $$ U^2=\sum_{i=1}^k X_i^2 $$ where $X$ is a random variable with normal standard distribution. What is the distribution of $X^2$...
1
vote
1answer
1k views

Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal $(\frac{1}{\sqrt{2\pi}})...
1
vote
1answer
36 views

Convolution with Gaussian, without distribution theory, part 2

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 $$ u(t,x)=...
5
votes
3answers
1k views

The probability density function of the ratio of two normal R.V.s

I'm looking for some help with this probability problem. Here's the question: Suppose that $X$ and $Y$ are independent standard normal random variables. Show that the probability density function ...
4
votes
1answer
508 views

Distribution of higher powers than 2 of a gaussian distribution

If $X \sim \mathcal{N}(0,1)$, then $X^2 \sim \chi^2(1)$. What about higher powers of $X$? I know that the Gamma Distribution is a generalization of the $\chi^2$ distribution, but I don't know how the ...
4
votes
2answers
502 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
2
votes
1answer
36 views

Convolution with Gaussian, without dstributioni theory, part 3

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 $$ u(t,x)=...
2
votes
1answer
2k views

Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent

If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent? What happens with the ...
0
votes
1answer
646 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim N(\...
0
votes
2answers
1k views

Indefinite integral of product of CDF and PDF of standard normal distribution

Is there a solution to: $\int ^\infty _x \Phi(z) \phi(z) dz$ where $z$ ~ $N(0,1)$ and $\Phi$ and $\phi$ refer to the CDF and PDF? Many thanks.
6
votes
1answer
106 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
5
votes
2answers
144 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 $$ u(t,x)=...
5
votes
2answers
1k views

Connection to Normal distribution

I've been working on finding the probability for the event, that the sum of $n$ independent random variables are less than $s$, when they are evenly distributed on $[0,1)$. I've used the law of total ...
4
votes
1answer
128 views

Point of maximal error in the normal approximation of the binomial distribution

I am sorry for the long question! Thanks for taking the time reading the question and for your answers! Context: Let $B_n\sim\text{Binomial(n,p)}$ be the number of successes in $n$ Bernoulli trials ...
3
votes
1answer
72 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
3
votes
2answers
225 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
2
votes
1answer
680 views

Sum of two independent normal distributed random variables

If $X_i$, $i =1,2$ are independent and have normal distribution with mean $0$ and variance $\sigma_i ^2$. Show that $X_1 + X_2$ has a normal distribution with mean $0$ and variance $\sigma_1^2 + \...
1
vote
2answers
812 views

Sum of two truncated gaussian

What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a ...
1
vote
2answers
15k views

Calculating mu and sigma (μ and σ) of a normal random variable

Let X be a normally distributed variable with unknown parameters μ and σ (sigma). If we know that P (X ≥ 75) = 0.7291 and P (X ≥ 83) = 0.7764. With the information given Is it possible to determine ...
1
vote
1answer
542 views

How to directly compute an integral which corresponds to the normal distribution

How does one directly (by finding primitive) compute an integral which corresponds to the normal distribution: $$\int_{a}^{b} e^{{-(x-a)^2}/{2s^2}} \,\mathrm{d}x$$
1
vote
3answers
70 views

Characteristic function of a square of normally distributed RV

Let's assume that $ X \sim \mathcal{N}(0,1) $. I'm supposed to compute the characteristic function of $ X^2 $. As far as I got is that the density of $ X^2 $ is $ g(y) = \frac{1}{2\sqrt{2\pi}} e^{-y/...
1
vote
3answers
35 views

Show that $Y\sim N(a+A\mu,AVA^T)$

Consider $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$. Show: If $a\in\mathbb{R}^m$ and $A$ is a $(m\times n)$-matrix with $\text{rang}(A)=m$ then $$ Y=a+AZ\sim N(a+A\mu,AVA^T). $$ My ...
1
vote
1answer
174 views

How to apply Gaussian kernel to smooth density of points on 2D (algorithmically)

I have a set of points on a 2D surface and need to build a heatmap. However, I also need to smooth out the density/distribution by applying some sort of kernel (Gaussian kernel, for example). I Know ...
0
votes
1answer
62 views

Normal distribution tail probability inequality

I am trying to show that $$P(X>t)\leq \frac{1}{2}e^\frac{-t^2}{2}$$ for $t>0$ where $X$ is a standard normal random variable. Perhaps this is simple. I have been starting with $$ \int_{t}^{\...
0
votes
1answer
55 views

Probability question of independent random varaibles

Let $X\sim \mathcal{N}(6,1)$ and $Y\sim\mathcal{N}(7,1)$ be two independent normal variables. Find $Pr(X>Y)$. the answer is $0.2389$ but I do not know how to do it.
0
votes
2answers
573 views

Proof that if $Z$ is standard normal, then Z^2 is distributed Chi-Square (1).

Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$. I want to use the method of moment generating functions, because I already understand the proof using the method of ...