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

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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?       &...
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1answer
32 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 normal],...
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1answer
53 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 = (B_{t_1},...
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0answers
64 views

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

Consider two random variables, $X$ and $Y$, with the following properties: $X\mid Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X\mid Y>y$ follow a normal distribution as well? If so, what ...
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2answers
37 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) $\dfrac{98}{...
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1answer
59 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 ...
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22 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 ...
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114 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
81 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 + \...
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1answer
25 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 \...
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1answer
38 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 ...
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1answer
49 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 $101$ ...
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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 ...
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1answer
41 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 ...
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1answer
134 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 ...
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0answers
40 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 ...
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1answer
69 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 ...
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1answer
61 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 ...
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1answer
171 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 ...
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2answers
60 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 ...
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1answer
37 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 (b). ...
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0answers
55 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 ...
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1answer
22 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 least ...
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1answer
74 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 ...
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1answer
19 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 + ...
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1answer
92 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
27 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 $P(\...
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0answers
28 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 ...
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1answer
65 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: $a)$ ...
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1answer
214 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} ...
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1answer
43 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} ...
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1answer
31 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) \...
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1answer
242 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 it'...
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1answer
184 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 ...
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1answer
128 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 ...
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2answers
30 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/ (...
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1answer
36 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 $$ u(t,x)=...
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1answer
36 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 $$ u(t,x)=...
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0answers
29 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 ...
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0answers
37 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 ...
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2answers
144 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 $$ u(t,x)=...
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1answer
19 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
30 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 ...
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2answers
312 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 $(S_k)_{k\geqslant1}$ ...
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2answers
98 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
45 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 ...
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1answer
28 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, ...
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1answer
57 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 don'...
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3answers
54 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: $$ P(\...
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2answers
60 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 $\boldsymbol\...