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

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2
votes
1answer
152 views

What is the distribution of empirical covariance between two independent normal distributions?

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
0
votes
4answers
690 views

Why this “interpolation” is correct?

Why does the interpolation in this image make any sense? Why won't I take $N(6.3) - N(6.1)$ or $N(6.2) - N(6.1)$? It is not linear at all.
0
votes
1answer
55 views

Parameter estimation with GMM

I have estimated the parameters of normal distribution with GMM and got the following results: $mean = -0.01168 , p-value = 0.83519, Sd = 1.77 , p-value = 0.00000.$ I'm bit confused in ...
1
vote
1answer
2k views

Variance for a product-normal distribution

I have two normally distributed random variables (zero mean), and I am interested in the distribution of their product; a normal product distribution. It's a strange distribution involving a delta ...
0
votes
1answer
176 views

Does an independent-increment Gaussian process necessarily have Gaussian increments?

Suppose a stochastic process is both independent-increment and Gaussian. Are all its increments Gaussian distributed? Thanks!
0
votes
1answer
114 views

Dividing data of a given distribution

Having a dataset with distribution $p$ (e.g. uniform or normal), if we divide the dataset into $n$ parts with equal size, is it valid to say that each part still has distribution $p$?
2
votes
2answers
291 views

What is the probability that two samples represent the same normal distribution?

Yes, it's a basic question. But, I have searched about 25 web pages for this and found only things that were irrelevant or incomprehensible. So I have indeed tried. My question is: I have two ...
-2
votes
1answer
73 views

Normal Distribution Stats

Let $$ X \sim N(65,20) $$ Find correct to $3$ Decimal Place the value of $x$ such that $Pr(X>x) = 0.43$
0
votes
1answer
177 views

Percentage point of Normal Distribution.

Let $$X \sim N(65,64) $$ Find the lower $2$% point for $X$; that is, find the value of $x$ such that $Pr(X<x) = 0.02 $ i know i need to do something like $\frac{X- 65 }8 = ... $ but not ...
0
votes
0answers
173 views

Statistics Question - Normal Distribution

The scores of a final exam have a Normal Distribution with mean $75$ and standard deviation 6. An independent sample size of $9$ is drawn from this distribution. The corresponding random variables are ...
3
votes
0answers
63 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
3
votes
0answers
84 views

Inverse of a sub-matrix

I have a multivariate Gaussian distribution with known $\mu$ and $\Sigma$. I want to evaluate it given a vector $x$. However, some of the attributes of this vector may be unknown, in which case I want ...
0
votes
1answer
110 views

How can I solve this integral?

How can I solve the following integral? $$\int_{-\infty}^\infty \prod_{i=1}^n \bigg( 1 - \Phi\left(\frac{c - \mu_i}{\sigma_i}\right) \bigg) \frac{1}{\sigma_Y}\phi \bigg(\frac{c-\mu_Y}{\sigma_Y} ...
0
votes
0answers
420 views

Numerical integration of 2-d Gaussian Distribution in MATLAB

I am looking for a really fast way to integrate numerically the 2-dimensional gaussian density with identity covariance matrix ...
0
votes
1answer
209 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 - $x(0.25)$.
4
votes
1answer
5k views

How to check if my dataset is normally distributed?

I have data sets (measurements) and I need to know if values are normally distributed. I would like to get this information programmatically in my application and not via plotting and checking it ...
0
votes
1answer
91 views

How to estimate parameters of a normal distribution?

Suppose one knew that 105 workers were evaluated by their boss. Such evaluation is distributed according to a normal distribution with mean $\mu$ and std. deviation $\sigma$. We also know that 20 ...
1
vote
1answer
180 views

Normal Distribution Identity

I have the following problem. I am reading the paper which uses this identity for a proof, but I can't see why or how to prove its true. Can you help me? \begin{align} \int_{x_{0}}^{\infty} e^{tx} ...
0
votes
1answer
45 views

question about normal distribution

I'm not being able to identify the mathematical pattern used answer this question: Suppose that the length of a speficic kind of snake may be modelled by a normal distribution with mean 50.8 (cm) ...
2
votes
1answer
139 views

Conditional distributions of the multivariate normal

Wikipedia gives details on the conditional distribution of the multivariate normal: If $\mu$ and $\Sigma$ are partitioned as follows $\boldsymbol\mu = \begin{bmatrix} \boldsymbol\mu_1 \\ ...
1
vote
2answers
70 views

Conditional Distributions and Probabilities

Suppose that $Y=A+\epsilon$ where $\epsilon$ is a RV and given some other random variable $\eta$ we have that: $\epsilon|\eta$ ~ $N(\rho\eta,\sigma^2)$ Suppose I was asked to find $Pr(Y=y|\eta)$ ...
0
votes
3answers
495 views

Integrating the pdf of a normal distribution

I need to find the distribution of $Y=X_1+X_2$ where both $X_1$ and $X_2$ are normally distributed with $(\mu,\sigma^2)$. So I'm looking for ...
1
vote
1answer
42 views

Distribution of proportions of each row cell

I'm trying to make sense of some data I have. Below is a simplified version of how data is structured. To get some context, the table shows the distribution of an investor's investments across ...
0
votes
1answer
189 views

moment generating functions by integration

Let X~N(0,1)m find the moment generating function of $X^2$ using integration techniques. I'm not sure exactly what this is asking me to do. Is $X^2$ just the pdf for the standard normal function ...
2
votes
0answers
49 views

Unknown result in probability theory relating CDF of any density to the CDF of normal distribution

There is apparently a result in probability theory saying: If $A(z)$ is any cumulative distribution function, $\alpha(t)$, the corresponding characteristic function and $\Phi(z) = ...
0
votes
1answer
31 views

closed form for $p(B_1>x>B_2)$ where $[B_1, B_2]'$ follows a bivariate lognormal dist?

Is there a closed form for $p(B_1>x>B_2)$ where $[B_1, B_2]'$ follows a bivariate lognormal dist: $$[B_1, B_2]' \sim \text{lognorm} (\boldsymbol \mu, \boldsymbol \Sigma)$$ where $\boldsymbol ...
2
votes
1answer
41 views

Multi-dimensional MLE Guassian

I wonder that what is the mu and sigma formula MLE(maximum likelihood estimates) for a 3 dimension guassian ? It is the same form as 1 and 2 dimension (+ 1 mu and sigma for the new vector) ?
0
votes
1answer
82 views

Filtering out noise from an approximate normal distribution

I'm dealing with sets of data that have distributions somewhat like this: ...
0
votes
0answers
58 views

What does taking the logarithm of a variable mean?

Question with regards to taking the logarithm of a variable (Statistics Question) Say you have a bar graph displaying data for an example "Cost of Computer Orders by the Population" and you are ...
2
votes
1answer
141 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 ...
0
votes
1answer
46 views

interpreting an expression involving two random variables

Consider a function $$g=E[\max(a+X,d+Y)]$$ where $a,d\in R$ and $X$ and $Y$ are independent and identically distributed standardized random variables with mean $\mu$, variance $\sigma^2$, continuous ...
0
votes
1answer
34 views

Computing expectation with n noisy sample?

Assume $\theta$~$N(0,\sigma^2)$, and we have $n$ realization of signals $s_i$, where $s_i$~$N(\theta,\sigma_i^2)$. Now the question is: what is $E[\theta|s_1,s_2,\dots,s_n]$? Thanks in advance.
0
votes
1answer
678 views

Application of Exponential Distribution

I'm in the process of developing a traffic simulation using Discrete Event Simulation approach. So I have the core stuff working but I need to have some sort of distribution to tell the simulation ...
2
votes
1answer
642 views

Problem with combination of discrete and continuous random variables

I'm pretty new to probability and a question is giving me some troubles. A binary information source produces $0$ and $1$ with equal probability. The output of the source, denoted as $X$, is ...
4
votes
0answers
75 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 ...
1
vote
1answer
2k views

Normal probability distribution with absolute value of X

Random variable X has a normal distribution N(30,5) find $P(|X| > 25)$ Having this I started to solve it normal way: $$P(|X| > 25) = 1 - P(|X| \le 25) $$ Now, normalize: $$1-P(|X| \le 25) = ...
0
votes
0answers
56 views

Is transfert theorem the best choice in this kind of exercise?

I am studying Probability theory and came to this exercise : Let $U,V$ be independent uniform random variables over $[0,1]$. Show that $X:=\cos(2\pi V)\sqrt{-2\ln U}$ and $Y:=\sin(2\pi V)\sqrt{-2\ln ...
0
votes
1answer
29 views

How to generate this vector?

I have two vectors $A$, $B$ each containing 10 random numbers from standard normal distributions. I want to generate another vector $C$ of $10$ numbers from standard distribution where ...
1
vote
2answers
83 views

Average error of two normally distributed measurements

There are two methods of measuring on object of length $x$. The error of the first method is normally distributed with a mean of 0 and standard deviation of $0.0056x$. The error made by the second ...
2
votes
1answer
178 views

Efficient calculation of the multivariate normal density function

The formula for the multivariate normal density function in the standard form contains $\Sigma^{-1}$ and the determinant of $\Sigma$, which are not very computationally friendly. Is it possible to ...
4
votes
1answer
254 views

Confusion related to integral of a Gaussian

I am a bit confused about calculating the integral of a Gaussian $$\int_{-\infty}^{\infty}e^{-x^{2}+bx+c}\:dx=\sqrt{\pi}e^{\frac{b^{2}}{4}+c}$$ Given above is the integral of a Gaussian. The ...
8
votes
2answers
310 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 ...
2
votes
1answer
124 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
1answer
521 views

Notation — What does “Gauss” brackets mean

In a paper I'm trying to understand, from a time series $x(1),x(2),\ldots,x(n)$ a new set of time series is created: $$x^m_k=x(m),x(m+k),x(m+2k),...,x\left(m+\left[ \frac{n-m}{k}\right]k\right) ...
1
vote
2answers
245 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
1
vote
1answer
64 views

Probability involving dependent normal variables

I have two independent, identically distributed normal random variables $X \sim N(0,\sigma^2)$, $Y \sim N(0,\sigma^2)$. I want to know $$Pr[X-Y>4\sigma \text{ and } X<3\sigma \text { and } ...
0
votes
1answer
574 views

equivalence between uniform and normal distribution

The principle of insufficient reason says that all outcomes are equiprobable when we have no knowledge to guess otherwise. I understand this and that this corresponds to uniform distribution. However, ...
1
vote
1answer
70 views

Distribution of largest sample from normal distribution.

Given $n$ independent random variables $X_i$ with normal distribution, mean $\mu$, variance $\sigma^2$, what is the distribution of $\max\limits_{i=1}^n(X_i)$ ? In particular I am interested in ...
3
votes
1answer
164 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 $$ ...
1
vote
1answer
239 views

The distribution when combining two samples together?

Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample ...