# Tagged Questions

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### Compound Gaussian distribution

Let $\mathbf{a},\mathbf{b}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbb{I})$ and let $A$ be the circulant matrix defined to have $\mathbf{a}$ as its first column. I'm trying to study the behaviour of ...
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### Minimum number of samples to take so that proportion of smokers in sample is within a certain threshold?

What is the minimum number of random samples that should be taken so that with probability at least 0.95, the proportion of smokers in the sample will not differ from the unknown population of smokers ...
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### Sum of dependent normal random variables

Let ${\bf X} =(X_1,\ldots,X_n)'$ be a vector of random variables that may be dependent and let ${\bf a}=(a_1,\ldots,a_n)'$ and ${\bf b}=(b_1,\ldots,b_n)'$ be nonrandom vectors with $a_i \neq 0$ and ...
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### $\int_{0}^{\infty}xe^{-x^2/2}dx= 1$?

$X \sim N(0, 1)$ $$E(|X|) = \frac1{\sqrt{2\pi}}\int_{-\infty}^{\infty}|x|e^{-x^2/2}dx= \frac{2}{\sqrt{2\pi}}\int_{0}^{\infty}xe^{-x^2/2}dx=\sqrt{\frac{2}{\pi}}$$ I don't understand how the last ...
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### Normal Distribution,standard deviation and probability question.

According a study, the duration of a match in World Cup is approximate normally distributed with the mean 111 minutes and standard deviation 5 minutes (including the break between the halves). ...
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### Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
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### Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
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### 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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### Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y$ have the following joint ...
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### Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
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### Marginalization of a paramter in Gaussian

If $\theta \sim N(\mu,\sigma_o^2)$ and $\mu \sim N(0, \sigma_1^2)$ what is the marginalized $P(\theta)$. Is it $N(0,\sigma_o^2+\sigma_1^2)$?
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### Normal Distribution and Probability on Excel

The size of fish in a lake follows a Normal Distribution with mean m = 1 lb 4 oz and standard deviation s = 3 oz . Fish that weigh less than 1 lb 9 oz must be released back into the lake. Bill ...
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### Sum of two truncated gaussian

What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a ...
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### Normal Distribution burnout… of lightbulbs.

Thank you for looking through this problem, much appreciated! I tried to work out the answer for a, but I got .2946 when the actual answer is .3085... How do I start this? By the way, I just want to ...
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### The Normal Distribution in measuring two towers…

I understand the explanation and the math behind the problem, all I am asking for is a quick explanation behind this. "Two instruments are used to measure the height, h, of a tower. The error made by ...
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### 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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### $\mathbb{P}(|X|<1,|Y|<2)$ 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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### 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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### Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= ...
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### 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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### How to compute the expected value of normal distribution over a finite interval.

The occurrence time of event A is normally distributed with mean $\mu=200$ and variance $\sigma^2=10^2$. That is, $f(A) \sim \mathcal{N}(200, 10^2)$. As known, the expected occurrence time of A can ...
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### Expectation of normal CDF with truncation

Suppose that $a$ and $T$ are given positive numbers. I would like to evaluate \begin{align*} \mathbb{E}\left[\Phi\left(aX\right)\mu\left(X+T\right) \right],\tag{1} \end{align*} where ...
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### Normal distribution without standard deviation given

The proportion of pink candies in a bag is supposed to be $50\%$. The filling machine is to be tested to see if it fills with the right proportion. A random sample of $50$ candies is made. The machine ...
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### When do normal distributions not occur?

I know that in many cases one can assume a normal distributed probability density. But what the situations when the distribution in non-normal. Some examples would be nice. For example, suppose ...
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### Estimation for expected value of a combination of two normal distributed random variables

I am struggeling to understand the proof of the following Lemma. Let $\epsilon_1$, $\epsilon_2$ be $\mathcal{N}(0,1)$ random variables. Then $\forall$ b $\geq$ 0 and $\forall$ c $\geq$ 1 there is an ...
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### Independent normal distributions

I found two theorems with a similar content and want to find out which one is true: Let $X,Y$ be normally distributed random variables and $X+Y$ is also normally distributed or $(X,Y)$ is ...
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### moment generating function of normal distribution

I know this question relates to the chi-squared distribution, but I think what the question wants me to do is somehow derive this distribution from the information given. I have a normally ...
If I have $(X,Y)$ with joint density $f(x,y)$ and $A$ is an invertible $2\times 2$ matrix, then for the random vector $(W,V)$ defined by: $$\begin{pmatrix} W\\ V \\ ... 1answer 48 views ### Calculation of distribution of a gaussian process Currently finishing the last year of PhD in statistics, we wonder if you could help us with the following. Let T = [0,1] and X = \left( X_{t}, t \in T \right) be a gaussian process with mean ... 1answer 52 views ### What's the pdf of Z=X^2 +2X if X is a standard normal? [closed] Le be X distributed as a standard normal. What is the density function of Z=X^2 +2X? Thanks for your help 1answer 28 views ### P-P plot and Q-Q plot How to draw P-P plot and Q-Q plot manually ? I have looked at different site and they explained in various way, such as one said for p-p plot in X-axis there is residual in ascending order and in ... 1answer 21 views ### limiting behavior of standard normal survivor function [duplicate] How do you show that \lim_{x\to \infty} 1-\Phi(x) \sim \phi(x)/x? In the previous, I'm using \Phi to refer to the standard normal CDF and \phi to refer to the standard normal pdf. Thanks!! 2answers 35 views ### Binomial and Normal Distribution Problem - Check solution Whooping cough is a highly contagious bacterial infection...About 80% of unvaccinated children who are exposed to whooping cough will develop the infection, as opposed to only about 5% of vaccinated ... 1answer 15 views ### Normal Distribution how N(x-x_n|0,\sigma^2) = N(x |x_n,\sigma^2)  I read an expression Could someone explain the step N(t-t_n|0,\sigma^2) = N(t | t_n,\sigma^2)  ? 0answers 17 views ### What is the minimum standard deviation for a normal PDF such that one tail is always larger than that of a second normal PDF (different means)? Say I have two weighted normal distributions,$$ f_1(x) = \frac{a}{2 \sigma_1} e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}} $$and$$ f_2(x) = \frac{1-a}{2 \sigma_2} e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} $$... 1answer 26 views ### Distribution of distance from 0 of gaussian point Suppose X_1,...,X_d\sim\mathcal{N}(0,1) are i.i.d.'s, each distributed normally around 0 with variation 1. It looks like \mathbb{E}\left(\sum X_i^2\right)=d. Why is that true? And how Y=\sum ... 1answer 144 views ### mean and variance normalization of vectors I have vectors x \in \mathbb{R}^n and I expect some multivariate normal distribution. I want to normalize the vectors in such a way that y = M (x - b) has mean zero (\operatorname{E}[Y] = 0) ... 1answer 40 views ### Standard deviation with multiple means and deviations The amounts of a certain mineral that can be produced in a day from mines 1, 2, and 3 are independent normal random variables with means equal to 80, 90, and 75 pounds, respectively, ... 1answer 22 views ### Exponential deviation with two x values I recently got interested in this topic of standard deviation. My TA did not have any time to go over this topic so I was trying to teach myself it recently. My TA said if he had more time he would ... 1answer 71 views ### Central Limit Theorem for uncorrelated (non-independent) but bounded random variables Given uncorrelated, discrete random variables X_i that are bounded, e.g., they can only take on values |X_i| \leq 4, then is there a form of the central limit theorem that one can apply to the ... 1answer 40 views ### How to find E[X|X>Y] Suppose X and Y are independent standard normal variable. I want to find E[X|X>Y]. I calculated that$$ f_{X>Y}(x) = 2\Phi(x)\phi(x) However I couldn't find the expected value using ...
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 ...