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

learn more… | top users | synonyms

1
vote
0answers
138 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
2answers
428 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
4answers
1k 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 ...
1
vote
1answer
98 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
311 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
186 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
51 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
61 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
292 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
87 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
61 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
152 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
70 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 ?
0
votes
1answer
70 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
188 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
151 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
99 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
74 views

Integral of an integral with variable limits

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