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

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1answer
32 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
26 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
47 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
29 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
15 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
49 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
31 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
49 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
36 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
26 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 ...
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0answers
15 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 ...
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1answer
19 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
34 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
18 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
75 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: ...
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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 ...
0
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0answers
27 views

Expectation of a Rayleigh-quotient-like form for normal random vectors

I have been trying to calculate or find a result for the expectation $$\mathbb{E} \left[ \frac{w^\top D^2 w}{1 + w^\top D w} \right] $$ where $$w \sim \mathcal{N}(0,I_N),$$ and $D \succeq 0$ is a ...
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0answers
19 views

Bivariate normal exercise - check my answer please

Similar to the question I asked before, with one subtle difference. If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are correlated with $\rho = 0.5$ then find: $a)$ the covariance ...
1
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1answer
23 views

Bivariate normal exercise - check please

I am trying to self learn some probability and wanted to ensure I was getting these exercises correct. If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are independent, then find: $a)$ ...
0
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1answer
58 views

expected value, random variable, piecewise function

I hope that you can help me with the following problem: Let $X \sim \mathcal N\left(\mu, \sigma^2\right)$ be a random variable. Define a new random variable $Y$ as: $$ Y = g(X) = \begin{cases} ...
1
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1answer
25 views

Finding the moments of normal variables

How to calculate the moments of a normal random variable with mean $\mu$ and variance $\sigma^2$? Using integration by parts we get the recurrence relation (calling $a_n = E(X^n)$) $$\begin{cases} ...
0
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1answer
25 views

Conditional probability distribution $p(A | A + B > C)$

Consider three independent normally distributed variables: A, B, C. How would you calculate the distribution $p(A | A + B > C)$? I know that the distribution $p(A + B | A + B > C) = p(A+B) ...
2
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1answer
86 views

Sum of truncated normal random variables

It's known that the sum of two independent normal random variables is itself normal. Does this hold when dealing with the sum of two truncated normal random variables? I've seen this question, but ...
0
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1answer
44 views

Normal Approximation to the Binomial question

I have a question I need help on: A supermarket manager samples n = 50 customers and if the true fraction of customers who dislike the policy is approximately .9, find the probability that the ...
1
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1answer
31 views

Probability that your return is positive for the week, given its distribution per year

You make an investment. Assume that returns are normally distributed with a mean return of .20 per year and a standard deviation of .10. Suppose you check on your returns once a week. What is the ...
0
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2answers
18 views

What is n value in a confidence interval

how large must n be if the length of the 99% CI is to be 40? the distribution is normal, sigma= 20. The book says that the answer is 7, but I keep getting 5.4 This is how I solved it: (X+Z(sigma/ ...
2
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1answer
33 views

Convolution with Gaussian, without dstributioni theory, part 3

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
1
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1answer
27 views

Convolution with Gaussian, without distribution theory, part 2

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
0
votes
0answers
26 views

Is this Integral transformation correct?

I have an Integral: $$ \int_{-\infty}^{-y_1} \Phi(y_2)d\Phi(x_1) $$ Here: $\Phi(y_2)$ is the Gaussian density function of variable '$y_2$' which has to be integrated w.r.t Gaussian density of ...
1
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0answers
21 views

Proving a process is a P Brownian Motion

Let $X_t = tW_{\frac{1}{t}} \forall t>0$ and $X_0 = 0$. I am trying to show that this process is a brownian motion under some measure P. I have shown that it is continuous and that it is ...
5
votes
2answers
124 views

Convolution with Gaussian, without distribution theory, part 1

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
0
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0answers
6 views

Parameters of gaussian distribution, which is generated using central limit theorem

In a software I am working on (sensor simulation), I needed to generate normally distributed noise for simulated sensor signals. I used the central limit theorem. I generated 20 random numbers and ...
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0answers
34 views

Finding the expected value of a A Gaussian voltage distributions?

A Gaussian voltage random variable $X$ has a mean of $ \over X $ = 0, and variance of $9$. The voltage $X$ is applied to a square-law, full-wave diode detector with a transfer characteristic $Y = ...
0
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0answers
17 views

Bivariate normal distribution: showing that linear combinations of joint Gaussians are Gaussian

Refreshing my stats, I wanted to learn how to derive the bivariate normal distribution, for which I found this source on Wolfram.com. On that website, the authors derive the joint probability ...
5
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0answers
107 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 ...
1
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2answers
73 views

Sum of a Normal and a Truncated Normal distribution

I have normal distribution $ N(\mu_1, \sigma_1)$ which shows the amount of demand in warehouse 1. The current amount of stock in the warehouse 1 is C. If the random demand is greater than C, it cannot ...
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0answers
25 views

Conditional PDF of multivariate normal distributions

Suppose that Y~$N\begin{pmatrix} 1\\ 2\end{pmatrix},\begin{pmatrix}2 & 1\\ 1 & 2 \end{pmatrix}$. How can I find the conditional PDF of $Y_1$ given that $Y_1+Y_2=3$?? I am given a hint to ...
0
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0answers
32 views

Proving chi-square distribution of estimator

There is something I can't get around my head. Let's suppose we havfe $ln x$ that is following a Normal distribution of parameter $lnx\rightarrow N(v;\theta )$ So we know that its estimator: $\hat ...
0
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1answer
26 views

Calculating standard deviation from a set of data

I'm trying to create a normal distribution of numbers between 0 and 100. I know that the mean = 28, and the only other information about the data is that there is a 10 % change that the number is 44, ...
0
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1answer
48 views

Why is Kurtosis of ND 3?

3 seems to be an important number when it comes to kurtosis. I see that it is often removed from the value entirely and this seems to be due to its being the kurtosis of the normal distribution. I ...
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3answers
37 views

Finding constant for CLT normal distribution

(This is from my textbook, but I don't understand their explanation. I've Googled around, but haven't found an answer that makes sense.) $$ \mu = 0, \sigma^2 = 1, n =16 $$ Find c such that: $$ ...
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vote
2answers
41 views

Determine $A$ such that $Q=X'AX$ has chi-squared distribution.

Let $\boldsymbol X\sim N_n(\boldsymbol\mu,\boldsymbol\Sigma)$, where $\boldsymbol\Sigma$ positive-definite. I am trying to determine, in general, what form $\boldsymbol A$ (one example is ...
0
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1answer
31 views

Complex circular symmetric Gaussian and real Gaussian

Circular symmetric complex Gaussian zero mean PDF is defined as : $$f(z)= \frac{1}{\pi^N||M||} e^{-z^*M^{-1}z} $$ where $M$ is hermitian semi positive definite, $z \in \mathbb{C}^{N \times 1}$ and ...
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1answer
28 views

Square of Normal distributed variable?

This is a quick question. If $X\sim\operatorname{Normal}$, is $X^2$ rayleigh distributed? I ask this question is because from wiki, it says $X^2$ is called the Chi-Squared Distribution with a degree ...
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0answers
46 views

Random numbers generator

If I know how to generate random numbers from Gaussian distribution (using Box-Muller method), how can I generate random numbers from distribution with pdf ...
0
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1answer
19 views

Limiting Behaviour of Root Mean Square Normal Random Variables - Related to Chi-Squared Distribution

Above is my question. I have done the first part - made hard work of it, albeit, but still, it's done. The next part is where I am stuck. Intuitively, it seems (to me!) like we should have $R_n ...
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3answers
28 views