Questions on the Gaussian, or normal probability distribution, and related topics.
35
votes
8answers
2k 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:
...
27
votes
2answers
2k 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 ...
10
votes
2answers
3k 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?
...
10
votes
1answer
465 views
Cool examples of the Central Limit Theorem in action
Sir Francis Galton has described the Central Limit Theorem as follows.
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of ...
7
votes
3answers
6k 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?
7
votes
2answers
459 views
Why don't we allow the canonical Gaussian distribution in infinite dimensional Hilbert space?
I'm looking at Gaussian distributions in infinite-dimensional Hilbert space, and the sources I've seen so far say that the covariance matrix has to be of trace class (i.e. the trace must be finite). ...
6
votes
3answers
195 views
How do I evaluate $\int \limits_{-\infty}^{a} e^{−t^2}dt$?
I know that
$$I \equiv \int \limits_{-\infty}^\infty e^{−t^2} \, dt=\sqrt{\pi},\text{ and }\int \limits_{-\infty}^0 e^{−t^2} \, dt=\frac{\sqrt{\pi}}{2}.$$
However, I don't understand if (or how) I ...
6
votes
1answer
458 views
Can the product of two non-independent Gaussians be Gaussian?
We recently discussed this: Is the product of two Gaussian random variables also a Gaussian? What was established was that in nontrivial cases (i.e., ruling out zero-variance Gaussians, which are ...
6
votes
2answers
119 views
Why should Gaussian noise have fractal dimension of 1.5?
In a paper I'm trying to understand, the following time series is generated as "simulated data":
$$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$
where $Z(j)$ is a Gaussian noise with ...
5
votes
1answer
105 views
How was the normal distribution derived?
Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
5
votes
1answer
2k views
X,Y are independent standard normal distributed then what is the distribution of $\frac{X}{X+Y}$
X, Y are independent standard normal random variables, what is the distribution of
$$ \frac{X}{X+Y} $$
Could anyone help me with this? Thanks.
I have worked the problem by multivariable ...
5
votes
1answer
42 views
Why does adding 3 random decimals in the range [-1,1] give a normal dist with std. dev 1?
I've used Math.random()*2-1+Math.random()*2-1+Math.random()*2-1 many times in the past to get normally-distributed random numbers with a standard deviation of 1. ...
5
votes
3answers
251 views
Compute probability of a particular ordering of normal random variables
There are $m$ normally distributed, independent random variables $N_1, \ldots, N_m$ with distinct means $\mu_1, \ldots \mu_m$ and standard deviations $\sigma_1, \ldots, \sigma_m$. Then, we get a ...
5
votes
0answers
64 views
What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?
If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed.
What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
5
votes
0answers
52 views
Volume of the intersection of ellipsoids
How do I compute the volume of the intersection of two n-dimensional ellipsoids?
Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
5
votes
1answer
233 views
Does this table fit the normal distribution?
The Pascal triangle can be described by the recurrence:
$P(n,1)=1, k>1: P(n,k) = P(n-i,k-1) + P(n-i,k)$
This well known triangle has the basic properties that the ratios of consecutive ...
4
votes
2answers
174 views
A simpler solution of the integral $\int_{x_1+\ldots+x_n \geq a} \exp\left[ -\pi \left(x_1^2+\ldots+x_n^2 \right)\right] dx_1\cdots dx_n $
I want to calculate the following integral
$$\int_{x_1+\ldots+x_n \geq a} \exp\left[ -\pi \left(x_1^2+\ldots+x_n^2 \right)\right]
dx_1\cdots dx_n, $$
as a function of $a$, in possibly the shortest ...
4
votes
4answers
107 views
Sampling from a $2$d normal with a given covariance matrix
How would one sample from the $2$-dimensional normal distribution with mean $0$ and covariance matrix $$\begin{bmatrix} a & b\\b & c \end{bmatrix}$$ given the ability to sample from the ...
4
votes
3answers
3k 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 ...
4
votes
4answers
969 views
Asymmetric Normal Probability Distribution
I'm looking for a continuous probability distribution a little bit like the normal distribution but asymmetric.
In my opinion this distribution applies to phenomenons related to response time in ...
4
votes
2answers
360 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
2answers
115 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 ...
4
votes
1answer
2k 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 ...
4
votes
3answers
142 views
7.7 standard deviations away from the mean?
I'm confused. I have a problem where I have to find the probability that x is below the z value 7.7. My z table only goes to z values of 3.4. How do I calculate this? These are the hints my teacher ...
4
votes
1answer
237 views
Generate a set of random numbers with a normal distribution
I am trying to generate a set of N random numbers where the set has a normal distribution.
I'm currently using a brute force approach:
Randomly select N numbers from a normal distribution.
Check ...
4
votes
2answers
62 views
Central Limit Theorem. How to apply to the task.
The research showed that the probabilities of 3, 4, 5, 6 and 7 cars broken on one day are 0.3, 0.4, 0.2, 0.08, 0.02 respectively. If 221 car broke in 50 days, does it show that more cars break than ...
4
votes
2answers
249 views
Property of gaussian integrals
Apologies if this has been asked before...
I came across the following relation:
if $$P(x_2, t_2 \mid x_1, t_1) = \frac{1}{\sqrt{2\pi\sigma^2(t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2\sigma^2(t_2-t_1)}}$$
...
4
votes
1answer
104 views
Expectations containing normal CDF
Suppose that $X\sim\mathcal{N}\left(0,1\right)$ (i.e., $X$ is a standard normal random variable) and $a,b,$ and $c$ are some real constant. Does any of the following expectations have a closed-form?
...
4
votes
1answer
166 views
Does the integral of PDF of multi-normal distribution over quarter planes have a closed form?
I am interested in finding a closed form solution (wich I suspect does not exist) to the following integral
$$\displaystyle \int _a^{\infty }\int _b^{\infty } \frac{\exp \left(-\frac{x^2+y^2-2 c x ...
4
votes
1answer
2k 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 ...
4
votes
3answers
222 views
Probability distribution function
I am trying to develop a function that will allow me to input a random number between 0 and 1 and receive a value. The idea is that the function has a range (for example, 0-100) with a median value of ...
4
votes
2answers
105 views
Minimizing the expectation over a set, wrt to the Gaussian measure
I have recently read a proof [1] where, at the last step, the authors use an inequality which basically amounts to a lower bound on
$\int_\mathbb{R} \mathbf{1}_A(x)|x| \phi(x)dx$, where $\phi$ is the ...
4
votes
1answer
33 views
What is the PDF of the Square Length of a Normally-Generated Vector?
Consider a vector $\mathbf{x}\in\mathbb{R}^n$, where each element in $\mathbf{x}$ is sampled independently from a normal distribution $\mathcal{N}(0,\sigma^2)$.
What is the probability density ...
4
votes
1answer
129 views
Fractional Part of Sum of Sequence of Independent Normal Random Variables
I'm trying to prove that if
$X_n$ iid normal
$S_n = \sum_1^n X_i$
$U_n=S_n-\lfloor S_n\rfloor$
then $U_n$ is asymptotically uniform in distribution. I've got no idea how to approach this, and it's ...
4
votes
1answer
99 views
how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)
I'd like to show the following equality (at least Mathematica claims it is an equality):
\begin{multline*}
\int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma ...
4
votes
0answers
63 views
The limit in law of a sequence of normal distributions is normal [duplicate]
Let $ \{ \xi_n \}_{n=1}^{\infty}$ be a sequence of normal random variables, where $ \xi_n\sim\mathcal{N}(\alpha_n, \sigma_n^2)$ and $\xi_n \overset{d}{\rightarrow} \xi$.
I need to prove, that $\xi$ is ...
4
votes
1answer
98 views
What quality of a distribution describes the “spikiness” of its density, and how do I get a good density plot of a spiky distribution?
I'm a programmer, not a math guy, so please answer in English. ;)
Suppose I have a multi-modal univariate distribution like:
.. . .. ........... .. . .. .
...
4
votes
0answers
59 views
Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?
I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants):
$$
\log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
4
votes
3answers
344 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 ...
3
votes
1answer
2k views
Why doesn't NORMSINV(RAND()) in Excel work as a standard normal random number generator?
I am looking for an easy way to generate random numbers from a standard normal distribution in Excel. I realize the best way is probably the Box–Muller method, ...
3
votes
3answers
623 views
Is a dot product between two independent multivariate Gaussian random variables also Gaussian random variable?
Let $x, z \sim N(0,I_p)$ be two independent multivariate Gaussian random variables. The question is whether the dot product $x'z$ is a Gaussian distributed variable.
My guess is that it is not. ...
3
votes
2answers
351 views
Probability of one normdist being greater than another
I have two independant normally distributed random variables.
X ~ N(657, 3)
Y ~ N(661, 2)
P(x > y) = ?
How do I calculate the probability of X being greater ...
3
votes
1answer
77 views
Conditional independence of differences between normal random variables
$X_1, X_2, X_3, X_4$ are independent, normally distributed random variables with different means and variances.
Let
$$ Y_1 = X_1 - X_2 \\
Y_2 = X_2 - X_3 \\
Y_3 = X_3 - X_4 \\ $$
Is it true that
$$ ...
3
votes
2answers
175 views
Two Sample Confidence Interval for Normal Distributions
Let's say I have two independent random samples $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$ from normal distributions with real, unknown means $\mu_x$ and $\mu_y$ and known standard deviations ...
3
votes
4answers
944 views
Calculate the expected value of $Y=e^X$ where $X \sim N(\mu, \sigma^2)$
I got a problem of calculating $E[e^X]$, where X follows a normal distribution $N(\mu, \sigma^2)$ of mean $\mu$ and standard deviation $\sigma$.
I still got no clue how to solve it. Assume $Y=e^X$. ...
3
votes
1answer
508 views
Multivariate Normal Difference Distribution
Since the distribution of a difference of two normally distributed variates X and Y with means and variances $(\mu_x,\sigma_x^2)$ and $(\mu_y,\sigma_y^2)$ respectively is given by another normal ...
3
votes
1answer
142 views
Expectation value of $1/x$
Given a random variable $x$ which is assumed to follow a Gaussian distribution
$x \sim N( \mu, \sigma^2 )$ and $x$ is further known to be positive, I am interested in the following expectation value: ...
3
votes
1answer
92 views
Is there a closed-form expression for the integral of this product of gaussian functions?
Considering:
$$f(x) = \frac{1}{\sigma_x\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x}{\sigma_x})^2}$$
$$g_i(x) = \frac{1}{\sigma_i\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{a_i+b_ix}{\sigma_i})^2}$$
Is there a ...
3
votes
1answer
58 views
Given that $X$ is normal, find the probability that $(X-10)^2 <12$
Suppose that $X$ is a random variable that has a normal distribution with mean = 5 and standard deviation = 10. Evaluate the following probabilities:
$\mathrm{Prob}((X-10)^2 < 12)$
3
votes
1answer
984 views
polar coordinates of Gaussian Distribution with non zero mean
I found that the polar coordinates of 2-dimensional Gaussian distribution with mean zero
$$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({x^2+y^2})/{2\sigma^2}\big) ...
