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

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1answer
37 views

Approximation in Normal distribution random variable

Let ${X_n : n \geq 1}$ be independent $\mathcal{N}(0,1)$ random variables. How do we get the following approximation?
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2answers
56 views

Normal distribution squared probability

Let $X_1,X_2,X_3,X_4$ be independent standard normal random variables and $Y=X^2_1+X^2_2+X^2_3+X^2_4$. Find the probability that $Y≤3$. Enter your answer as a decimal and make sure that at least $10$ ...
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3answers
65 views

Computing standard deviation of discrete normal distribution

I used below pseudocode to generate a discrete normal distribution over 101 points. ...
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2answers
49 views

What is the probability that a Chi-square distribution lies within 2 standard deviation of its mean?

Here I have an 8 degrees of freedom Chi-square distribution function $f(x)$ So by definition, $E(X)=8, Var(X)=2*8=16$. (Please guide me if this is wrong. We just started this chapter and there's ...
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3answers
47 views

integrating $A^2=\frac{1}{2\pi}\int^\infty_{-\infty}\int^\infty_{-\infty}e^{-\frac{y^2+z^2}{2}}dydz$

When proving that $$\int^{\infty}_{-\infty}\frac{1}{\sqrt{2\pi}\sigma}{e^{-\frac{1}{2}({\frac{x-\mu}{\sigma})}^2}}dx=1$$ and I faced a problem, ...
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1answer
154 views

Find the standard deviation of $ \frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$

Given $\frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$ as a normal distribution PDF with mean $\mu$, I'd like to solve for the std deviation in terms ...
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1answer
278 views

How do I prove Poisson appraches Normal distribution

I want to prove why the mean and variance of a $\operatorname{Poisson}(\lambda)$, is different when the time index approaches infinite (it's approximated by the mean and variance of a Normal). For ...
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3answers
523 views

Integrating the pdf of a normal distribution

I need to find the distribution of $Y=X_1+X_2$ where both $X_1$ and $X_2$ are normally distributed with $(\mu,\sigma^2)$. So I'm looking for ...
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2answers
129 views

$k$-dimensional normal distribution function

If $f(\vec{x})$ is a vector of normal distribution function and assuming that $\sigma$ is same in all dimension, can we say that $$ f(\vec{x}) = \prod_{1\leq i \leq k} \frac{1}{\sqrt{2\pi\sigma^2}} ...
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2answers
655 views

Mean euclidean distance for normal-random coords

What is the mean euclidean distane between two points on the plane which coordinates are normally distributed? I'm assuming this would be $\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int ...
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1answer
34 views

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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1answer
60 views

integral with pdf of a gaussian

$$ I = \int_{0}^{\infty} x \phi(x) dx $$ where $\phi(x)$ is the pdf of a normal distribution. Here I read that: If $X = \mu + \sigma U$ with $U$ a std normal, $$ I = E[\mu + \sigma U; mu + \sigma ...
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1answer
42 views

Find the probability that the average of X and Z is greater than Y. Where X, Z, and Y are normal RVs.

Here is the exact statement: Suppose X,Y , and Z are independent random variables. X is a normal random variable with mean 5 and variance 16, Y is a normal random variable with mean 7 and variance ...
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2answers
55 views

Central Limit Theorem Application on Multivariate Normal

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. What is the distribution of $\overline{X} = \frac{1}{3}(X_1+X_2+X_3)$? I don't quite understand how to ...
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2answers
57 views

MGF/ expectation Gaussian Random Variables

I am stuck with something that seems easy but i cannot recall how to figure it out? Let $G_1$ and $G_2$ be two standard gaussian random variables with mean $0$ and variance $1$. Then how to calculate ...
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1answer
158 views

Probability: normal distribution and standard normal random variable

Let $X$ follows the normal distribution $N(1,9)$. Find $\text{(a)}$ $P(X\le1.4).$ $\text{(b)}$ $P(X\le-1.22).$ $\text{(c)}$ Hence find $P(-1.22\le X\le1.4).$ For $\text{(a)}$, is ...
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1answer
2k views

How to calculate the probability of a normal distribution with unknown mean and unknown variance?

How do you calculate the probability of a normal distribution with unknown mean and unknown variance? If a problem stated, for example, that 15% of the time sales are more than 15,000 and 20% of the ...
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2answers
30 views

Error in solving for raw score; incorrect formula used?

According to a study of how long a person is willing to wait for their flight, it is found that the mean time a person is willing to wait is 5.2 hours with a standard deviation of 1.1 hours. Consider ...
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2answers
249 views

How do you determine the sample size of a normal distribution?

I am presented with the question: The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Assume that the thickness is ...
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1answer
134 views

what is the pattern in the distribution of divisors.

I made a table that shows the number of divisors for each number less than 500, and i think that there is a pattern, for example when there is a spike in the number of divisors the surrounding numbers ...
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1answer
109 views

Normal random Vector

Question: Prove that linear functions of the form $\bar{y}=\bar{b}+\mathrm{B}\bar{x}$ are normal random vectors provided that $\bar{x}$ is a normal random vector. Find $E(\bar{y})$ and $V(\bar{y})$. ...
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1answer
153 views

special matrix in terms of its covariance matrix

How can we find a matrix $S\in \mathcal{M}_{n,n}$ and $Z\in \mathcal{M}_{n,m}$ whose $n$ entries of the $i^{th}$ column $Z_i$ are correlated $Z_i \sim \mathcal{N}(0,S)$ where $S \in \mathcal{M}_{n,n}$ ...
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2answers
44 views

Algorithm for integral of standard distribution

I need help in producing random data that follows standard distribution. Since it is to be used in a computer application, I would prefer an algorithm before a table. So, this is what I need. The ...
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2answers
161 views

What is the equation of the expectation of the product of two normally distributed multivariate random variables?

Given two multivariate random variables $\mathbf{x} \sim N(\hat{\mathbf{x}}, Q)$, $\mathbf{y}\sim N(\hat{\mathbf{y}}, R)$ which are not independent but are correlated with covariance matrix $C$ and a ...
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1answer
186 views

Calculating P(A>B), where A and B are normal distribution

In the problem we have that A ~ N(7, 11/60) and B ~ N(7.3, 7/20) and the question is what is the probability that A gives a higher value that B. Since the textbook we have for the course doesn't ...
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1answer
40 views

Understanding edge correction with a 2nd order polynomial in Gaussian filter

I am trying to understand the following code from ImageJ: http://pastebin.com/tXfhNxqf The problem: When computing the gaussian kernel we use the gaussian function $$ f(x) = e^{-\dfrac{x^2}{2 ...
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1answer
18 views

Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
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1answer
21 views

Plotting Normal Distribution using Excel

I was trying to experiment some stuff (scaling issues and hypothesis testing) with normal distribution. While doing so, I found out that : NORM.S.DIST(0, FALSE), which takes Z-value, returns prob. ...
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2answers
48 views

Central Limit Theorem not valid?

According to Central Limit Theorem (CLT), the mean of any i.i.d. sample is Normal distributed (taking $n\rightarrow\infty$ samples). Let $X_i\sim U(a,b)$. Then $\bar{X}\sim N$ by CLT. But as we ...
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1answer
21 views

Normal Random Variables

Let Z1 and Z2 be independent standard normal random variables. What is the probability that the minimum of Z1 and Z2 will be greater than 1.0? How do I go about this when I have no values? Is the ...
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2answers
35 views

Given N-distribution, calculate expected value and Var of a function

$X_{{i}}=X_{{1}}...X_{{n}}$ is an iid. random variable with the distribution $N \left( {\frac {\alpha}{\beta}},{\frac {{\alpha}^{4}}{{\beta}^{4}}} \right) $ I would like to calculate the expectation ...
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2answers
35 views

Sum of maximum of two correlated normal random sequences

Let $x_{1},x_{2},\cdots,x_{n}$ and $y_{1},y_{2},\cdots,y_{n}$ be correlated normal random variables the covariance between two arbitrary random variables is known. In other words, let ...
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1answer
50 views

Why is this multivariate $3\sigma$ ellipse rotated?

While reading this answer, I clicked on the provided link to this Wikipedia page. The main article image shows the PDF of a 2D multivariate normally distributed system: In the image, the $3\sigma$ ...
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2answers
40 views

“Show experimentally” that for large $N$, $X$ appears to be normally distributed.

I'm a bit confused about the following problem: Let $X$ be the random variable $$X = \frac{X_1+X_2+...+X_N}{\sqrt{N}}$$ where $X_k$ is the outcome from the $kth$ flip of a fair coin where heads ...
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1answer
16 views

$\mathbb{P}(|X|<1,|Y|<2)$ 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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1answer
28 views

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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2answers
14 views

Determining the marginal distribution

Consider $X=(X_1,\ldots,X_n)^T\sim\mathcal{N}(\mu,V)$. Show that then $X_i\sim\mathcal{N}(\mu_i,V_{ii})$ for all $1\leqslant i\leqslant n$. Good day! Ok, I have to determine the marginal ...
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1answer
47 views

Likelihood ratio critical region

Let $X_{1},..,X_{n}$ be a random sample from a normal distribution with mean ${\theta}$ and variance 1. We wish to test $H_{0}:{\theta}=0$ vs $H_{1}:{\theta}{\neq}0$. Write down likelihood ratio ...
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1answer
44 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, ...
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1answer
20 views

Standard normal RV probability

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 ...
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1answer
30 views

upper and lower bounds of $E[X|X>x]$

I am trying to find tight upper and lower bounds for $E[X | X > x]$ where $X$ follows a standard normal random variable. After calculations I found that $$ E [X|X>x] = ...
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1answer
61 views

How Moment generation function of Gaussian R.V. can be divided into this one?

I know the moment generating function of Gaussian random variable is $$E\{e^{rx}\}=\int^{+\infty}_{-\infty}e^{rx}f(x)dx=e^{mr+r^2\sigma^2/2}$$ where $f(x)$ is PDF of Gaussian R.V., $m$ is mean value ...
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1answer
29 views

Two different ways to calculate the probability for a negative value - are they equivalent?

I want to calculate $$ P(Z\leq-1.8) $$ My math book teaches this one: $$ F_Z(0) - F_1(1.8) = 0.5 - F_1(1.8) $$ This makes sense. But what about the following? Does it also make sense, is it ...
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1answer
62 views

How can we derive expectation of two dependent normal distribution?

$\mathbf{X}$ and $\mathbf{Y}$ are each dependent normal random variable, then how can we derive like this one? $$\mathbf{E}\{e^{\mathbf{X}}e^{\mathbf{Y}}\}$$ I know the each first moment is ...
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2answers
97 views

Adding two normal distribution

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. And Suppose that $Z \sim N(1, 2^2)$ and is independent of all $X_i$. Define $Z_i = Z + X_i$ for $i = 1, ...
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2answers
72 views

how to prove $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$

Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $N(0,\frac{a^3}{3})$. means $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
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1answer
63 views

What is the effect of the variance on a sequence of cumulative product?

We randomly draw numbers from a normal distribution with mean equals $mu$ and variance equals $var$. We draw the values: $x_1, x_2, x_3, x_4, ...$ Then, we construct a sequence made of the ...
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2answers
104 views

Multivariate normal distribution from invertable covariance matrix

I want to generate a random vector with $\mathcal{N}(0, C)$ distribution, i.e. normal distribution with $0$ mean and given covariance matrix $C$. $C$ is not invertible (singular). Here it's written: ...
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1answer
147 views

Forumla for finding conditional variance

I need to find the conditional variance $\mathop{\mathrm{Var}}(X_1|(X_2+X_3))$, given that $X_1\sim N(0,1)$ and $X_2+X_3\sim N(0,2+2\gamma)$. The covariance between X1, X2+X3 is $\rho$. From this ...
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1answer
77 views

Conditional Normal Distribution of Mice

The weights of a population of mice fed a certain diet follow a normal distribution with mean $\mu=100$ grams and standard deviation $\sigma=20$ grams. A random sample of $8$ such mice is taken. Let ...