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

learn more… | top users | synonyms

0
votes
1answer
13 views

How to calculate covariance of X and Y given joint probability

$X$ and $Y$ are dependent variables both normally distributed as $N(\mu-const, \sigma^2)$. I don't know what the joint distribution is. I know that when $const = 0$, then the joint probability ...
1
vote
3answers
35 views

Normal distribution exercise!

If a technician does not encounters any hardware problems, the time he requires to assemble a computer follows a normal distribution with a mean of $30$ minutes and a standard deviation of $3$ ...
0
votes
0answers
11 views

Normal distribution problem using R

The elongation of a steel bar under a particular load has been established to be normally distributed with a mean of $\mu = 0.05$ and a standard deviation of $ \sigma = 0.01$. Find the probability ...
0
votes
2answers
22 views

How to calculate the joint probability from two normal distributions

I have two random variables $X$ and $Y$ both normally distributed as $N(\mu, \sigma^2)$ (they have the same distribution). $X$ and $Y$ are dependent. They are defined from other random variables A, B ...
0
votes
1answer
10 views

Optimize distributions for low mean, high variance

Assume a context with $N$ approximately normal distributions where a lower mean implies a 'better' distribution and a high variance or high standard deviation implies a 'better' distribution as well. ...
2
votes
1answer
38 views

Show that $W$ is a Gaussian process

I have the following problem: I want to prove that the vector $(W(1_{[t_0,t_1]}),...,W(1_{[t_{n-1},t_n]}))$ is normally distributed with mean $0$ and covariance matrix ...
1
vote
1answer
29 views

Statistics - Exponential distribution

There are $n$ machines. Each has durability given by exponential distribution with $EX = 10$. If a dead machine is replaced with new one immediately, find minimal $n$ so we can say with $P = 0.99$ ...
1
vote
0answers
20 views

Standard normal distribution hazard rate

Is the hazard rate of the standard normal distribution convex? Can you give a reference?
-1
votes
1answer
40 views

Statistics - normal distribution problem

Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. If $P(X > 3) = 0.8413$ find $P((X+Y)/2 > 3)$. The result must be exact number, so normal distribution parameters are ...
0
votes
1answer
469 views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
0
votes
2answers
25 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
1
vote
1answer
197 views

What proportion are above x of sample size n where X ~ N(0,1) Homework

I have a homework question that I'm not quiet sure of. It follows as so: Consider a random variable $X$ that has a standard normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$. ...
0
votes
2answers
361 views

normal distribution using Z - finding probability between 2 numbers

I am wanting to find the probability of the following: SD = 20 Mean = 100 P(85 < X < 117) i have found the z values for both: P(X>85) : X-u/o = 85-100/20 Z = -0.75 and found the ...
1
vote
1answer
19 views

Cholesky decomposition and variance

Well, I've been reading about simulating correlated data and I've come across Cholesky decomposition. Everything seemed clear until I found a couple of posts on this site and Cross-Validated that ...
1
vote
0answers
19 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
0
votes
2answers
371 views

Standard deviation of less than one

How would I find the approximate percentage of values within a standard deviation of less than one on the normal model? Chebyshev's rule is only used when the standard deviation is greater than or ...
0
votes
1answer
418 views

Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= ...
5
votes
1answer
64 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where ...
2
votes
3answers
98 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
3
votes
1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
2
votes
1answer
534 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
4
votes
1answer
29 views

Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
0
votes
1answer
14 views

Probability of return with 7% error

I have a problem understanding the answer of the following problem: A recent audit by the IRS of the returns she prepared indicated that an error was made on 7% of the returns she prepared last ...
5
votes
1answer
66 views

Checking the Lindeberg condition (central limit theorem)

Problem. Let $W_1, W_2,...$ be independent and identically distributed random variables such that $E(W_1)=0$ and $\sigma^2 := V(W_1) \in (0,\infty)$. Let $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j ...
1
vote
4answers
52 views

The normal distribution is a common model of randomness

Can someone please comment/elaborate on the statement: "The normal distribution is a common model of randomness." I would like to understand it more deeply. Source: Perhaps someone can point me ...
5
votes
1answer
109 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
1
vote
1answer
36 views

Statistics- Finding Probability

A local lawn service has determined the average time it takes to mow an average residential yard is thirty-five minutes. If mowing times are independent and constant, what is the probability it will ...
1
vote
1answer
31 views

Question regarding the derivation of the distribution of $(n-1)S^2/(\sigma^2)$

I will quote from my statistics manual: "Consider a random sample $x_1,x_2...x_n$ taken from a population with distribution $N(\mu,\sigma^2)$, whose average $\mu$ is unknown; [through the central ...
8
votes
2answers
207 views

Joint distribution of the signs of the partial sums of independent standard normal random variables

Consider some i.i.d. standard normal random variables. What is the joint distribution of the signs of their partial sums? More formally, define a sequence of random variables ...
2
votes
2answers
29 views

Gaussian function constant

Why are 1D gaussians defined as $$F(x;\sigma^2) = e^{\frac{-x^2}{2\sigma^2}}$$ for a probability function (after computing the gaussian integral): $$p_F(x;\sigma^2) = ...
0
votes
1answer
28 views

Probability of agreeing to do some work depending on the payment

I am looking for several options of modeling the probability of people agreeing to do some work depending on the price/payment. The payment can only range between $p_1$ and $p_2$, $(p_1 < p_2)$. I ...
9
votes
0answers
92 views

Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
2
votes
1answer
45 views

Prove that the product of 2 vectors Normally distributed converges for large dimensions to the full zero matrix

Let $\mathbf{x}, \mathbf{y}$ $\in C^{M \times 1}$ are two i.i.d. vectors with distribution $\mathcal{CN(0,1)}$. How we can prove by the strong law of large numbers that: $\lim_{M\rightarrow \infty} ...
0
votes
1answer
18 views

Expectation of normal and log normal distribution

Let $X \sim N(\mu_x, \sigma_x^2)$ and $Y\sim N(\mu_y, \sigma_y^2)$, with correlation $\rho$. How do I find $$E[Xe^Y]$$? I tried a bunch of things without result. I'm also interested in "general" ...
0
votes
0answers
11 views

pseudo-Wishart distribution with shifted rows

I have a problem and I don't know where to start finding a solution. The problem is that I have a vector of i.i.d normal random variables such that, ...
0
votes
1answer
21 views

Properties of a distribution function

I'm having trouble understanding the properties of a distribution function. My book only gives these short rules. http://www.pixhost.org/show/2720/28297379_2015-06-22-15-27-44.jpg My professor said ...
0
votes
1answer
39 views

Correlation coefficients of X and Y [closed]

I was wandering if anybody could help me with the following question. I am fairly new to correlation coefficients and was attempting to tackle this question but was unsure how to do so? Thanks.
1
vote
1answer
51 views

To find $\sigma$ of a normal distribution

Given $X \sim \mathcal{N}( n, \sigma^2)$. The question told me $\mathbb{P}(X\lt 3) = \mathbb{P}(X\gt 7)$ So I found $n$ which is $5$. I'm also given $2\mathbb{P}(X\lt 2) = \mathbb{P}(X\lt 8)$. ...
0
votes
1answer
370 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
0
votes
2answers
41 views

Explaining the form of the Gaussian measure

The Gaussian density $\mu(dx)=e^{-x^2/2}\ dx$ is fundamental in probability theory. Does anyone have a (non-computational) heuristic why this function should be special? (By non-computational, I mean ...
2
votes
1answer
551 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
0
votes
0answers
22 views

How do I add uncertainties?

I have a gas for which we are continuously measuring composition with an online instrument. This composition is then being used to calculate some properties (dewpoint), and I want to estimate the ...
1
vote
1answer
438 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
0answers
10 views

Dice probability normal distribution

You roll a dice 1000 times. Calculate the probability you roll a six between 150 and 200 times. I understand how you calculate this with the binomial distribution: $$ = Binomialcdf(1000, 1/6, 200) - ...
0
votes
1answer
38 views

Normal distribution for bags of coal produced from a machine.

A machine is used to bag coal, the mass of coal delivered per bag being normally distributed with mean 55 Kg and standard deviation 1.25 Kg. Given two filled bags chosen at random calculate the ...
0
votes
1answer
23 views

A convergent sequence of normal random variables

Say $\{X_n\}$ is a sequence of normal random variables with means $0$ and variances $\sigma_n^2$. Also suppose that $X_n\to X$ (everywhere) and $\sigma_n^2\to \sigma^2.$ Then, using characteristic ...
0
votes
0answers
29 views

Equally spaced numbers with interval

I've this very silly doubt in getting equally spaced numbers with fixed interval from a set of numbers. Let's say I have a set of points, if I want maximum N points from this set equally distanced, ...
1
vote
1answer
24 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
2
votes
1answer
21 views

Expectation of cumulative distribution function of a standard normal distributed random variable

Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. The find the expectation of $\Phi(X)$. I ...
0
votes
0answers
29 views

Probability involving Normal Distribution

Two instruments are used to measure the height, h, of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056h. The error made by the ...