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

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1answer
27 views

Would the joint distribution of Normal Random Variable and the distribution of a X bar from the same sample be bivariate Normal?

I know this question is somewhat redundant... but here goes: My text asserts that the joint distribution of $$X_1=N(\theta, 1)\text{ and } \bar X = N(\theta, \frac 1n)$$ is Bivariate normal with ...
1
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1answer
91 views

The probability that the ratio of two independent standard normal variables is less than $1$

Let the independent random variables $X,Y\sim N(0,1)$. Prove that $P(X/Y < 1) = 3/4. $ Could anyone help me prove this analytically? Thanks. Progress: My first thought was to integrate the joint ...
2
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1answer
50 views

Normal Distribution Probability with known mean and variance

I believe I am quite close to solving this, but I would just like to double check some of these answers. Two species have different size toes. Lengths of toes of species X is normal distributed with ...
3
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1answer
27 views

Basically Normal Dist question

I'm a little rusty on my probability and would appreciate any help. I think I have done the bulk of the work already anyway, but my question is: If $X \sim LN(1,2)$ find $P(X>1)$ $X$ being ...
1
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1answer
30 views

multivariate normal moment derivation

I am having trouble deriving the mean for a multivariate normal for $\mathbf{x} \sim \mathbb{N}(\mathbf{m},\Sigma)$: $$ \mathbb{E}[\mathbf{x}]= \int_{R^d} \mathbf{x} ...
4
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1answer
116 views

Bayesian Updating with 1 Signal but 2 Unknowns

Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 different values {${\alpha_1, \alpha_2}$} such that $\alpha = \alpha_1$ with probability $p_1$ and $\beta$ is ...
0
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1answer
29 views

Distribution with density $x^2\operatorname{exp}\{-x^2/2\}$

I came across the probability distribution with density $$ f(x)=\sqrt{\frac{2}{\pi}}\,x^2\,\mathrm{e}^{-\frac{x^2}{2}},\quad x\geqslant 0. $$ Is this distribution known under a certain name? I only ...
0
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1answer
22 views

The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution

Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ...
2
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1answer
103 views

Simple calculations of mean, standard deviation, and probability

You are a successful entrepreneur that has developed a new sustainable product that is manufactured through a standard production process. As part of this process, the product goes through quality ...
0
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2answers
54 views

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve? a) $P(0 < Z < 2.17)$ b) $P(-2.50 < Z < ...
1
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0answers
18 views

Normal distribution tables - right or left?

Are the probabilities in normal distribution tables given typically to the right or left of the $Z$ score? One such text I am reading says to the right. However, in my lecturer's exercises, I ...
0
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0answers
19 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
0
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1answer
52 views

Generate a uniform distribution from n coin flips

I'm making a computer game and I've reduced the problem into something simple: How can I show the player the number of heads he "tossed" given some number of coins = n? Naive expected value is ...
0
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0answers
30 views

Moments of Multivariate Normal Distributions

I have two questions. Suppose we have two multivariate normal distributions $X \sim N(\mu,\Sigma)$ and $Y\sim N(c\mu,\Sigma)$ where $0<c<1$ is a constant, $\mu$ is a vector and $\Sigma$ is a ...
2
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1answer
31 views

Find probability given a binomial and a normal distribution

$X$~$Bin(n,p),Y_n$~$N(μ,\sigma^2)$ Where X is the number of trials taking place, and $Y_n$ is the amount of time the $n$th trial takes (independent of other trials). $Z$ is a new random variable ...
1
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2answers
21 views

Calculating a normal distribution with a sample size?

the sample of $n=25$ is what is throwing me off. I have no clue what to do with it. Given a normal distribution with $\mu=101, \sigma=25$, and given you select of $n=25$ $A.)$ $P(\overline{X} ...
1
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0answers
40 views

Approximating the integral of the exponential of a quadratic function

An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where $c=b^2/(4a)+\ln(-a/\pi)/2$ and ...
0
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0answers
39 views

How to construct a two sided confidence interval?

A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected and the diameter is measured. The resulting data are shown below. 5.21 5.28 5.29 5.27 ...
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0answers
21 views

Finding the conditional distribution of a normal RV given another normal RV

I'm having trouble with this question from a past qualifying exam: Question Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. ...
1
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0answers
26 views

Can a mixture of normals be a constant?

Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane?     ...
0
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1answer
29 views

Square Matrix Algebra - help please!

I am stuck on a problem in matrix algebra and I would be happy if someone could help me. Given a square matrix with dimensions "p" given that $\textbf{x}$ $\sim$ N($\mu,\Sigma$) [multivariate ...
1
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1answer
43 views

How to interpret the covariance matrix of Brownian motion

I'm reading Bernt Oksendal's "Stochastic Differential Equations". It says, Brownian motion $B_t$ is Gaussian Process, i.e. for all $0 \leq t1 \leq \cdots \leq t_k$ the random variable $Z = ...
1
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0answers
47 views

If $X|Y$ and $Y$ are both normal, is $X|Y>y$ normal as well?

Consider two random variables, $X$ and $Y$, with the following properties: $X|Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X|Y>y$ follow a normal distribution as well? If so, what are its ...
5
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2answers
33 views

correlation between $\sum_{i=1}^{98}X_i$ and $\sum_{i=3}^{100}X_i$

Let $X_1,...,X_{100}$ be iid $N(0,1)$ random variables. The correlation between $\sum\limits_{i=1}^{98}X_i$ and $\sum\limits_{i=3}^{100}X_i$ is equal to (A) $0$ (B) $\dfrac{96}{98}$ (C) ...
0
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1answer
54 views

Central Limit Theorem sample vs population

I need help in the setup of this problem. I'm sure that I'm making this far more complicated than what it actually needs to be. "An anthropologist wishes to estimate the average height of men for a ...
0
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0answers
20 views

If I approximate a Bionimial distribution with a Normal Distribution am I still allowed to use Binomial's equation for Variance?

If I approximate a Binomial distribution with a Normal Distribution am I still allowed to use Binomial 's equation for Variance? So am I still allowed to use this: $Var(x) = np(1 − p)$ While still ...
0
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0answers
67 views

Expected value of a function of truncated normal

I need to find the expected value of the following type of an expression: $$\mathbb{E}[\frac{1-\alpha}{1-\alpha-\frac{X}{\beta}}]$$ where $\alpha$ and $\beta$ are constants and $X$ is a random ...
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0answers
50 views

Question Relating to the Central Limit Theorem

I have the following question: Suppose $X_1,X_2, \ldots, X_{12}$ are identical independent uniform random variables on $[0,1]$. Let the sample mean(lets call it $m$) = $\frac{1}{12}(X_1 + X_2 + ...
0
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0answers
20 views

Distribution of unknown, given system of equations

Suppose we have an unknown real $x$. We want to give an approximation of $x$ by measuring the distance between $s_i$ and $x$, for $i = 1,2,3$. The position of each $s_i$ is distributed with mean $p_i$ ...
0
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1answer
22 views

Calculate $E(X^2)$ of random variable $X$ ~ $N(3,4)$

I need to find $E(X^2)$ of random variable $X$ ~ $N(3,4)$. I can use the simple way: $E(X^2) = \int_{-\infty}^{\infty} x^2 \cdot f(x) dx$, in this case $f(x) = normal \space distribution ...
0
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0answers
20 views

Poisson distribution with normal informative priors

I'm Jia, a student of economics and finance. I was wondering if someone could help in understanding this problem. I've just started to attend a new course "Financial and nonlinear econometrics" and ...
0
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1answer
34 views

Distribution of two independent standard normals

Suppose that $X$ and $Y$ are distributed as independent Standard Normals. Find the distribution of $(X-Y, X+Y)$. Isn't the case for $X-Y$ elementary? Since they are both standard normals, this ...
0
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1answer
35 views

Chi distribution and sample variance

Suppose that the height (in cm) of randomly selected male is distributed according to normal distribution with parameters $\mu = 175$ and $\sigma = 5$. We pick a simple random sample of size ...
0
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3answers
49 views

Expected value of random variable $X$ ~ $N(170, 25)$

Here's a question: Person's height in CMs is a random variable $X$ ~ $N(170, 25)$. Door's height is $180$ cm. What is the expected value of number of people that can enter the door until the first ...
3
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1answer
33 views

Seminorms in distribution theory are norms, right?

In distribution theory the seminorms that you use there $p_m( \phi) := \max_{|\alpha| \le m} \sup_{x \in \Omega} |(\partial^{\alpha}(\phi) (x)|, \phi \in C_c^{\infty}(\Omega)$. Those guys are norms ...
1
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1answer
43 views

Random Gaussian variable raised to arbitrary power

Given $x$ that follows a distribution $P(x)=e^{\frac{-x^2}{2\sigma^2}}$ i.e. a random Gaussian variable, can I say anything about the distribution of $x^n$ for fixed $n$? Specifically, is there ever ...
1
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0answers
17 views

sampling from a multivariate guassian: intuition behind using cholesky decomposition

I'm trying to understand how sampling from a multivariate gaussian works and why the cholesky decomposition is a way to do it. Let's say we have a 25 dimensional multivariate with a 25x25 covariance ...
1
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1answer
50 views

Smallest n to align sample mean with population mean

There's a question in my book that I just do not understand. This is it in its entirety: Let $ \bar{X} $ be the sample mean of a random sample of size $ n $ from a normal distribution with a variance ...
0
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1answer
37 views

On the notation of normal distribution

I saw in the Finnish matriculation examination solutions the sentence If $X$ has the distribution $N(100,15)$, $Z=\frac{X-100}{15}$ has the distribution $N(0,1)$. How one can memorize this? I mean ...
2
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1answer
61 views

Sample standard deviation and population standard deviation

The average temperature of a particular tropical island is normally distributed with a mean of 74 degrees and a variance of 9 degrees (a) If a random sample of 16 days has been taken, what is the ...
4
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2answers
41 views

How to set up normal approximation for binomial

In a particular school, 25% of first grade students do not enjoy reading. 22% of second graders do not enjoy reading. A random sample is taken of 100 first grade students, and another independent ...
0
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0answers
29 views

Sum of a Normal and a Truncated Normal distribution on Mathematica

I asked a question about the "Sum of a Normal and a Truncated Normal distribution" about 11 days ago, and someone helped me (I appreciate his\her help a lot). I tried to do same procedure as he\she ...
0
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0answers
17 views

Statistics, distributions and graphing

Lets say that you have data containing one variable - the waiting time between each car passing from 1200 hours... so you have something like 23, 54, 26, 8, 2, 59 etc what is the best way to analyse ...
1
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1answer
20 views

Solving for an unknown $\mu$ in a probability problem involving normal random variables.

(a): $P[X < 355] = P[Z < \frac{355 - 360}{4}] = P[Z < -1.25] = 1 - \Phi[1.25] = .1056$. Part (a) is simple, but I included it because I was not sure if I should somehow use it to solve ...
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0answers
38 views

How to obtain the pdf of this transformation of normal random variables?

Given the following: Let $r = ab + n$, where $a, b,$ and $n$ are independent zero-mean Gaussian random variables with variances $\sigma_a$, $\sigma_b$, and $\sigma_n$, respectively. Find the MAP ...
0
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1answer
19 views

I'm unsure of the setup for this probability question from the society of actuaries

The answer is 0.223584. Here is my attempt: Company A: $\mu = 10000\\ \sigma = 2000\\ \text{40% chance of at least one claim}$ Company B: $\mu = 9000\\ \sigma = 2000\\ \text{30% chance of at ...
1
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1answer
41 views

Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.

My question is: do you know any examples when $X$ and $Y$ are both normally distributed, but the two dimensional vector $(X,Y)$ is not? I found some example in book, but I don't understand it. The ...
2
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1answer
16 views

Linear transformations in normal distributions

I am still a bit new to this topic, and was wondering if someone could check my work, it is a short exercise. Find the distribution of $X = \mu + N(0,1)$ If we let $Z \sim N(0,1)$ then $X = \mu + ...
-1
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1answer
79 views

find distribution of hypothesis testing? [closed]

Suppose $x_1,x_2,...,x_{20}$ is a random sample from a normal population with mean = 0 and variance $ \sigma ^2 $. I want to test the hypothesis $H_0: \sigma ^2 \geq 4$ against the alternative $H_1: ...
0
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0answers
22 views

a question in Stat. aboout chi-square & standard normal

Assume $U$~$\chi^2(5)$, $V$~$\chi^2(9)$, $Z$~$N(0,1)$, U, V, Z are mutually independent, calculate: a. $P(Z > 0.611V^\frac{1}{2})$ b. $P(\frac{U}{V} < 1.933)$ c. Find a $c$ such that ...