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

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2
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1answer
2k views

Linear 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 ...
2
votes
1answer
282 views

Iteratively Updating a Normal Distribution

Is there a way to update a normal distribution when given new data points without knowing the original data points? What is the minimum information that would need to be known? For example, if I know ...
2
votes
3answers
10k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
1
vote
1answer
73 views

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in ...
1
vote
1answer
88 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 ...
1
vote
1answer
9k views

What is the expectation of $ X^2$ where $ X$ is distributed normally?

I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal. Thanks
0
votes
1answer
118 views

Why is this multivariate $3\sigma$ ellipse rotated?

While reading this answer, I clicked on the provided link to this Wikipedia page. The main article image shows the PDF of a 2D multivariate normally distributed system: In the image, the $3\sigma$ ...
0
votes
1answer
120 views

Normal random Vector

Question: Prove that linear functions of the form $\bar{y}=\bar{b}+\mathrm{B}\bar{x}$ are normal random vectors provided that $\bar{x}$ is a normal random vector. Find $E(\bar{y})$ and $V(\bar{y})$. ...
6
votes
1answer
102 views

An interesting inequality about the cdf of the normal distribution

When approaching this other question I came out with the inequality: $$\frac{1}{4+x^2}e^{-x^2/2} \leq\Phi(x)\Phi(-x)\leq \frac{1}{4}e^{-x^2/2},\tag{1}$$ where $\Phi(x)$ is the cdf of the standard ...
4
votes
2answers
626 views

Convolute exponential with a gaussian

I have data measuring an exponential decay that is convoluted by a gaussian response function. I have the measured shape of the gaussian, and want an analytical expression for the exponential ...
4
votes
3answers
283 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 ...
3
votes
2answers
101 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 ...
3
votes
0answers
148 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
3
votes
2answers
2k views

Normal Distribution, The “Y” Value

Guys I am having trouble with the standard normal distribution. http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm We know the X values run from approx $-\infty$ to $+\infty$ but ...
3
votes
2answers
529 views

P.D.F. of independent/dependent Uniform R.V.'s

I am trying to solve this: Consider a stick of length 1. You break the stick in two random places, X and Y. a. Define the individual probability distribution functions of the breaking ...
2
votes
1answer
42 views

If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...
2
votes
2answers
84 views

Find distribution $Y=X^2$

X~N(0,1). Find distribution $Y=X^2$ Can someone help me? I have no idea how to do it. I could try to start like this: $F_Y(t)=P(X^2<t)=P(-\sqrt(t)<X<\sqrt{t})$
2
votes
1answer
40 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
2
votes
1answer
476 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 $Y = \dfrac{1}{X}$
2
votes
1answer
397 views

Brownian Bridge as a Gaussian Process

Let $B=\{B_t:t\geq 0\}$ be a standard Brownian motion. Define the Brownian brige $X=\{X_t:t\geq0\}$ as $$ X_t=B_t-tB_1\quad t\in[0,1] $$ Show that $X$ is (i) Gaussian and find its (ii) mean and (iii) ...
2
votes
1answer
151 views

Conditional Expectations (Mainly an integral question)

Let $X_1$ and $X_2$ be two Random variables with a standard normal distribution, and the two variables are independent. Find $E[X_1|X_1>X_2]$ My answer is far. If we knew $X_2$, then the answer ...
2
votes
1answer
153 views

Techniques for evaluating probability integral

Consider the integral of a normal distribution: $$\int_a^b f(x)\,\mathrm d x=c $$ and a second integral for the expected value: $$ \int_a^b x\cdot f(x)\,\mathrm dx $$ Since you know the first ...
2
votes
3answers
2k views

$3\sigma$ rule for multivariate normal distribution

I was wondering if the $3\sigma$ rule that holds for 1D normal distribution also holds for multivariate normal distribution?
2
votes
1answer
375 views

Combining 1D normal distributions into a 2D distribution

First of all, apologies for my poor terminology - I have a particular problem which I understand in own terms, but I am having difficulty in applying the mathematics in the correct manner. My problem ...
2
votes
1answer
3k views

Definite integral of Normal Distribution [duplicate]

Possible Duplicate: How to directly compute an integral which corresponds to the normal distribution Is there any approximate solution for the following definite integral of normal ...
2
votes
4answers
6k views

Convolution of two Gaussians is a Gaussian

I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the ...
1
vote
1answer
540 views

Probability: Normal Distribution

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability $0.95$. Approximate the probability (by a normal distribution) that at most $10$ of the ...
1
vote
1answer
37 views

Probability , Geometric and Gaussian

So,I'm good at the questions which require the understanding of basic formulaes , but this one my prof said needs me to think (for the first one)'geometrically'=Stumped. Please Help! The second is an ...
1
vote
0answers
237 views

Almost sure convergence of maximum in a sequence of Gaussian random variables

Let $X_1, X_2,\ldots,X_n$ be an i.i.d. sequence of standard Gaussian variables and $M_n=\max(X_1, X_2,\ldots,X_n)$. I am trying to understand the mechanics of the proof of almost sure convergence ...
1
vote
1answer
106 views

Does $0$ correlation imply independence for marginally normal distributions?

Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$. Can someone give a hint why this is true ?
1
vote
2answers
596 views

The correlation between two normal distribution

Let $X$ have the $N(0,1)$ distribution and let $a>0$, show that the random variable $Y$ given by $$Y=\begin{cases} X & \text{if }|X|<a\\[5pt] -X &\text{if }|X|\geq a\; \end{cases}$$ has ...
1
vote
1answer
146 views

Scaling of a multivariate normal

We know that if a variable $X$ is iid from a $N(\mu,\sigma^2)$, the distribution of $X+b$ is $N(\mu+b,\sigma^2)$ If we scale the $X$ by a scaling factor $k$, the new distribution will be ...
1
vote
2answers
72 views

Statistical Problem (part 2)

Following my question I found another problem. Having the same data from the other question: There are 2 melon stores. The melon weights follow a normal distribution. Store A -> μ = ...
1
vote
2answers
353 views

Definite integral of cdf of the form $\Phi(\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}})$

Any solution for the following definite integaral? Here $\Phi(x)$ represents the cumulative distributive function of standard normal distribution ...
1
vote
1answer
214 views

Multivariate Normal Distribution

Is is true to say that k-dimensional Normal distribution is equivalent to the multiplication of k 1-dimensional Normal distributions if variance is equal in all dimensions?
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0answers
25 views

Why we can use normal distribution to approximate binomial distribution when n is large enough?

Prove why we can use normal distribution to approximate binomial distribution when n is large enough. Hint: Try to read something on bernoull ...
0
votes
1answer
41 views

How to find expected value of a portion of the normal distribution?

$X\sim N(67,4)$ What's the expected value of the portion of the curve $(X>72)$? I tried to use the definition of expected value ($\int xf(x) \mathrm{d}x$), but my integral was far too complicated ...
0
votes
1answer
31 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
0
votes
1answer
105 views

how to generate Normally distributed random number?

I am looking for a function that can generate Normally distributed random numbers. I came to know about bux-muller transform but I didn't understood it completely what it is doing. Thus it would be ...
0
votes
1answer
67 views

What is the effect of the variance on a sequence of cumulative product?

We randomly draw numbers from a normal distribution with mean equals $mu$ and variance equals $var$. We draw the values: $x_1, x_2, x_3, x_4, ...$ Then, we construct a sequence made of the ...
0
votes
1answer
527 views

Normal approximation to the log-normal distribution

Intuitively, it seems that a lognormal distribution with a tiny $\sigma/\mu$ ratio might look quite a bit like a normal distribution. Can this be formalized in any way (e.g., by stating upper bounds ...
0
votes
1answer
381 views

Conditional expectation on components of gaussian vector

I think I got the definition of the conditional expectation now, but I'm still having some problems with actual calculations... Let $(X,Y,Z)$ be a real gaussian vector. X and Y centered and ...
0
votes
2answers
117 views

Normal distribution probability problem.

There are lots of salmon in a pond and their length (in centimeters) obeys normal distribution $N(70, 5.4^2)$. You and your friend go fishing and decide to continue fishing until both of you catch at ...
0
votes
1answer
214 views

Ratio of dependent chi squared random variables

Suppose that $X=v'A_1v$ and $Y=v'A_2v$, where $A_i$ are symmetric matrices and $v$ a multivariate normal vector with covariance $V$, are chi squared distributed each with its own degrees of freedom. ...
0
votes
0answers
69 views

iterative transform of standard normal random variable

Given a discrete series of random variable $n(i)$ that each element follows the standard normal distribution $N(0,1)$, another series is defined iteratively as: $$u(i+1)=au(i)+bn(i)$$ where ...
0
votes
1answer
122 views

how to do this integral: $ \int_{0}^{\infty} \int_{0}^{\infty} x y \phi(x, y) dx dy$

how to do this integral: $$ \int_{0}^{\infty} \int_{0}^{\infty} x y \phi(x, y) dx dy$$ where $ \phi(x,y)$ is a general pdf of bivariate normal distribution, that is: $$\phi(x,y) = ...
0
votes
1answer
153 views

special matrix in terms of its covariance matrix

How can we find a matrix $S\in \mathcal{M}_{n,n}$ and $Z\in \mathcal{M}_{n,m}$ whose $n$ entries of the $i^{th}$ column $Z_i$ are correlated $Z_i \sim \mathcal{N}(0,S)$ where $S \in \mathcal{M}_{n,n}$ ...
0
votes
1answer
105 views

What is the distribution of an unconditioned random variable knowing the conditional distribution?

I have two random variables $X$ and $Y$. I know that $Y$ can be approximated by a $N(\mu_1,\sigma_1^2)$ distribution (in particular $Y$ is not negative) and I also know that $X|Y \sim N(a+bY,c+dY)$ ...
0
votes
1answer
335 views

How to count $n$th percentile from normally distributed random variable?

I have normally distributed random variable $X\sim \mathcal N(100,225)$. How to count $n$th percentile? In my case I need lower quartile - $x(0.25)$.
0
votes
1answer
228 views

Calculating P(A>B), where A and B are normal distribution

In the problem we have that A ~ N(7, 11/60) and B ~ N(7.3, 7/20) and the question is what is the probability that A gives a higher value that B. Since the textbook we have for the course doesn't ...