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

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-5
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0answers
19 views

How to Solve A Normal Distribution problem? [on hold]

The teacher gives a quiz. The scores were normally distributed with a mean of 30 and a standard deviation of 2.5. If there are 25 students in the class, what is the expected number of students to ...
0
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0answers
11 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 ...
0
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0answers
11 views

Normally Distributed Summation of Random Variable

Suppose that at ABC Company there is only one customer representative. Let N Bin(10, 0.6) be the number of customers requiring service in one hour, and Si N(10, 5) be the service time (in minutes) ...
1
vote
0answers
15 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. ...
-2
votes
0answers
19 views

The Empirical Rule [on hold]

For the year 2003, the SAT test scores were normally distributed with a mean of 1050 and a standard deviation of 150. If 1495 SAT scores are in the top 0.14%, estimate how many students took the SAT ...
1
vote
0answers
21 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
votes
0answers
12 views

Expectation of Multivariate Normal - help please

I am struggling to show why an equation is true. Help would be very much appreciated Given that x $\sim$ N($\textbf{m}$, $\Sigma$) [multivariate normal], how could you show that: E[($\textbf{x}$ - ...
0
votes
1answer
21 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
vote
1answer
32 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
vote
0answers
39 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
votes
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
votes
1answer
46 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
votes
0answers
17 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
votes
0answers
44 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 ...
1
vote
0answers
20 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
votes
0answers
10 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
votes
1answer
20 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
votes
0answers
19 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
votes
1answer
31 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
votes
1answer
17 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
votes
3answers
40 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
votes
1answer
27 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
vote
1answer
17 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
vote
0answers
13 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
vote
1answer
48 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
votes
1answer
21 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
votes
1answer
37 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
votes
2answers
32 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
votes
0answers
23 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
votes
0answers
10 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
vote
1answer
14 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 ...
1
vote
0answers
30 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
votes
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
vote
1answer
37 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
votes
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 + ...
0
votes
1answer
61 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: ...
-1
votes
1answer
64 views

one random instance with 4 elements and includes mean?

if we select one random instance with 4 elements from normal distribution, and we show minimum value among this instances with a, and show maximum value among this ...
0
votes
0answers
21 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
votes
0answers
20 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 ...
1
vote
0answers
18 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
vote
1answer
20 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
votes
1answer
34 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
vote
1answer
22 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
votes
1answer
24 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
votes
1answer
59 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
votes
1answer
32 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
vote
1answer
20 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
votes
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
votes
1answer
30 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
vote
1answer
26 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 $$ ...