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

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$\mathbb{P}(|X|<1,|Y|<2)$ When $X,Y$ Are I.I.D. Standard Normal

Calculate $\mathbb{P}(|X|<1,|Y|<2)$ when $X,Y$ are i.i.d. standard normal r.v.s. I think the answer is simply $$(\Phi(1)-\Phi(-1))(\Phi(2)-\Phi(-2)).$$ Is this correct? Thanks.
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22 views

An IB Math HL question on normally distributed random variable.

Some Background: Tim goes to a popular restaurant that does not take any reservations for tables. It has been determined that the waiting times for a table are normally distributed with a mean of ...
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2answers
14 views

Determining the marginal distribution

Consider $X=(X_1,\ldots,X_n)^T\sim\mathcal{N}(\mu,V)$. Show that then $X_i\sim\mathcal{N}(\mu_i,V_{ii})$ for all $1\leqslant i\leqslant n$. Good day! Ok, I have to determine the marginal ...
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1answer
46 views

Likelihood ratio critical region

Let $X_{1},..,X_{n}$ be a random sample from a normal distribution with mean ${\theta}$ and variance 1. We wish to test $H_{0}:{\theta}=0$ vs $H_{1}:{\theta}{\neq}0$. Write down likelihood ratio ...
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37 views

Standard deviation with multiple means and deviations

The amounts of a certain mineral that can be produced in a day from mines $1$, $2$, and $3$ are independent normal random variables with means equal to $80$, $90$, and $75$ pounds, respectively, ...
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20 views

Standard normal RV probability

Z is a standard normal random variable I need to find $P(|Z|<.95)$, find c such that $P(|Z|<c)$, and given that X is a RV with mean 3 and standard deviation 16, find $P(X>3.84)$ I am just ...
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28 views

upper and lower bounds of $E[X|X>x]$

I am trying to find tight upper and lower bounds for $E[X | X > x]$ where $X$ follows a standard normal random variable. After calculations I found that $$ E [X|X>x] = ...
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60 views

How Moment generation function of Gaussian R.V. can be divided into this one?

I know the moment generating function of Gaussian random variable is $$E\{e^{rx}\}=\int^{+\infty}_{-\infty}e^{rx}f(x)dx=e^{mr+r^2\sigma^2/2}$$ where $f(x)$ is PDF of Gaussian R.V., $m$ is mean value ...
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29 views

Two different ways to calculate the probability for a negative value - are they equivalent?

I want to calculate $$ P(Z\leq-1.8) $$ My math book teaches this one: $$ F_Z(0) - F_1(1.8) = 0.5 - F_1(1.8) $$ This makes sense. But what about the following? Does it also make sense, is it ...
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61 views

How can we derive expectation of two dependent normal distribution?

$\mathbf{X}$ and $\mathbf{Y}$ are each dependent normal random variable, then how can we derive like this one? $$\mathbf{E}\{e^{\mathbf{X}}e^{\mathbf{Y}}\}$$ I know the each first moment is ...
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420 views

'normally distributed random numbers' vs 'uniformly distributed random number'?

what is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)
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91 views

Adding two normal distribution

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. And Suppose that $Z \sim N(1, 2^2)$ and is independent of all $X_i$. Define $Z_i = Z + X_i$ for $i = 1, ...
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2answers
72 views

how to prove $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$

Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $N(0,\frac{a^3}{3})$. means $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
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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 ...
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2answers
100 views

Multivariate normal distribution from invertable covariance matrix

I want to generate a random vector with $\mathcal{N}(0, C)$ distribution, i.e. normal distribution with $0$ mean and given covariance matrix $C$. $C$ is not invertible (singular). Here it's written: ...
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146 views

Forumla for finding conditional variance

I need to find the conditional variance $\mathop{\mathrm{Var}}(X_1|(X_2+X_3))$, given that $X_1\sim N(0,1)$ and $X_2+X_3\sim N(0,2+2\gamma)$. The covariance between X1, X2+X3 is $\rho$. From this ...
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75 views

Conditional Normal Distribution of Mice

The weights of a population of mice fed a certain diet follow a normal distribution with mean $\mu=100$ grams and standard deviation $\sigma=20$ grams. A random sample of $8$ such mice is taken. Let ...
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103 views

Joint distribution of two marginal normal random variables

Question: Suppose we have: \begin{align*} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim N\left(\begin{bmatrix} 6 \\ 3 \end{bmatrix}, \begin{bmatrix} 12 & 3 \\ 3 & 2 \end{bmatrix} \right) ...
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200 views

Probability of a normal distribution; more than, less than confusion.

You are interested in finding how many hours a person is willing to wait for a plane. It is found that the time people are willing to wait has a $μ = 5.2$ and a $σ = 1.1$. What is the probability a ...
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1answer
81 views

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need?

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need? I may not be using the correct terminology so here's a graph: Based on this, if you ...
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1answer
519 views

How to merge two Gaussians

I have two multivariate Gaussians each defined by mean vectors and Covariance matrices (diagonal matrices). I want to merge them to have a single Gaussian i.e. I assume there is only one Gaussian but ...
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2answers
62 views

Noise pdf Gaussian

Why the probability distribution function of the noise in a channel is Gaussian (normal distribution)? Intuitive discussion is appreciated.
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371 views

Why do we use a $z$-test rather than a $t$-test when estimating an appropriate sample size?

I'm kinda puzzled on one point. In our stat class, we are taught to use the Student $t$ distribution to find confidence intervals for normally distributed data, as blindly using the normal ...
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2answers
571 views

Indefinite integral of product of CDF and PDF of standard normal distribution

Is there a solution to: $\int ^\infty _x \Phi(z) \phi(z) dz$ where $z$ ~ $N(0,1)$ and $\Phi$ and $\phi$ refer to the CDF and PDF? Many thanks.
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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 - $x(0.25)$.
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1answer
966 views

Cumulative distribution function determine the random variable

I don't know that determine is the right word, but I try to explain. What I need to understand. :) So.. We know's that if a function fit this conditions: Monotonically non-decreasing for each of its ...
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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 ...
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2answers
996 views

Adjusting mean and standard deviation

There's a set of 8 bags with the following weights in grams given: 1013, 997, 1013, 1013, 1004, 985, 991, 997 The mean is 1001.625, unbiased standard deviation is 10.86. I have the following ...
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3answers
116 views

Mixture of two mirror-image Gaussians

Suppose we are given a set of points $(x_i, y_i)\in\mathbb{R}^2$ and are told that they are drawn from a normal (Gaussian) distribution. It is a simple matter in that case to find the mean ...
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1answer
86 views

$\alpha$-stable distributions and Gauss distribution tails differences.

I know that in a $\alpha$-stable distribution we have: $$ \lim_{x\rightarrow +\infty}f(x,\alpha,\beta)\sim -\alpha \gamma^\alpha \frac{\Gamma(\alpha)}{\pi}sin(\frac{\pi ...
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384 views

Evenly distributed normal distribution on surface of a unit sphere

Hello I am using Mathematica code to to generate NPoints of randomly generated points normally distributed around one point on the surface of a sphere using Return[Table[ ...
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1k views

How to integrate $\int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx$?

How can I integrate, $$ \int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx $$ and what's the result w.r.t the Gaussian function's p.d.f $p(x)$ and c.d.f $\phi(x)$? Thanks!
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3answers
451 views

What does it mean for two random variables to have bivariate normal distribution?

The following is Sheldon Ross's definition: We say that the random variables $X,Y$ have a bivariate normal distribution if, for some constants $\mu_x,\mu_y,\sigma_x>0,\sigma_y>0, ...
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25 views

Area under Normal Distribution Curve

What is the formula that determines the Z-score table? More specifically, what formula can be used the equate the area underneath the normal distribution curve, without using the table?
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3answers
38 views

Question about normal approximation and variance

This isn't so much a question about getting a right answer as much as it's about understanding a mathematical concept, but I will give you the problem that spawned it: An analysis of data shows that ...
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1answer
13 views

Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
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1answer
16 views

How to determine the distribution of $U:=(X,Y,Z)$?

I've got a question concerning the distribution of a multi dimensional random variable. I know that $X$ and $Y$ and $Z$ are each normal distributed with certain expectations and variances. ...
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1answer
30 views

Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $E[||x||_2]$, $x $~$ ...
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1answer
7 views

Bivariate Normal Probability

Assume we have a large data set of PSAT and SAT scores with bivariate normal distribution with $\rho = 0.6$. The mean and SD of the PSAT scores are (respectively) $1200$ and $100$. The mean and SD ...
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1answer
10 views

Distribute range of score among objects

I need some help with the following. I have 10 or X amount of subjects with a rating and would like to distribute a score of 1 to 5 between them based on their rating. The subject with the highest ...
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1answer
21 views

Model going from Normal to Log-Normal

I'm getting in a real mess at the moment over something I think is very simple, as well as the wording/terminology. I have a model - $\ln(Y(x))=a+b\ln(x)+\epsilon, ...
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2answers
75 views

Distribution of mean of Normal distribution

Suppose $X\sim N(\mu,\sigma)$. I want to find the following probability $P[\mu \ge \theta |x= \theta -c]$ for $c>0$. In another word, I saw a sample of Normal distribution, $x$, and know that it ...
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1answer
35 views

Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
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37 views

Independent variables, normal distribution, pdf

I have independent variables $ X_1, X_2,\ldots,X_n $ with normal distribution on range $ [0,1] $ . Next, variables $ Z_i $ are created according to this formula $ Z_i = - \frac{1}{\lambda} \ln(1-X_i) ...
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2answers
29 views

Numerical precision of product of probabilities (normal CDF)

I'm trying to calculate $\prod_k{p_k}$ where $p_k$ are (potentially) very high probabilities of independent, zero-mean, standard normal random variables and $k>100$. However, I'm running into ...
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1answer
26 views

Distribution of distance from 0 of gaussian point

Suppose $X_1,...,X_d\sim\mathcal{N}(0,1)$ are i.i.d.'s, each distributed normally around 0 with variation 1. It looks like $\mathbb{E}\left(\sum X_i^2\right)=d$. Why is that true? And how $Y=\sum ...
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1answer
51 views

Bringing a density in a normal distribution form

Because I do not want to exaggerate this thread Show that $E(Y\mid X=x)$ is a linear function in $x$ I continue my special problem here. In order to make the setting clear I'll give some information. ...
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2answers
60 views

Painful? Moment Generating Function

Part 1 Let $X$ be a random variable with the p.d.f. $f(x)=\frac{1}{4\pi}e^{\frac{-x^2}{4}}$, compute the MGF of $X$. So I know I want ...
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77 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
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1answer
13 views

Geometric Sequence with Normal Distribution Problem

Given: The running time (in seconds) of an algorithm on a data set is approximately normally distributed with mean 3 and variance 0.25. a. What is the probability that the running time of a run ...