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

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2
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3answers
1k 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
359 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
3answers
8k 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 ...
2
votes
1answer
2k 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
5k 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
36 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
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0answers
161 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
95 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
497 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
112 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
70 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
334 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
195 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?
0
votes
1answer
68 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
63 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
312 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
99 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
0answers
63 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
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
83 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
214 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
183 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 ...
0
votes
2answers
108 views

Normal distribution, statistical problem

Before proceeding to the question, bear in english is not my native language and therefore technical terms may be wrong. So, I'm trying to solve the old exam question, and I have different results ...
0
votes
1answer
1k views

How to apply Central Limit Theorem to Uniform Distribution to generate Normal Distrubution?

Suppose I have a simple uniform continuous "unit" distribution X: $$\begin{align*} \forall y \in \mathbb{R} \implies \\ y < 0 : & P(X < y) = 0 \\ y \in [0,1] : & P(X < y) = y \\ ...
-1
votes
2answers
88 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...