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

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Generate a set of random numbers with an average evenly distributed between two given values

1) I generate 1000 random numbers between 0 and 10 and take the average. If I do the above action "many" times the resulting average values will be a normal distribution over 0 to 10. Correct? What ...
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28 views

Why a normal distribution would not give a good approximation to the distribution of marks

An examination is marked out of $100$. It is taken by a large number of candidates. The mean mark, for all candidates, is $72.1$ and the standard deviation is $15.2$. Give a reason why a normal ...
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30 views

How to calculate inverse cumulative distribution using a table?

I need help with this: $$P(X\geq a)=1-F_X(a)=1-\Phi\left(\frac{a-70}{8}\right)=0.25$$ When $X\sim N(70,64)$. I know that it should be: $(a-70)/8 = 0.6745$ How do I get $0.6745$ From $Z$ table? I ...
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38 views

$\lim X_n = 0$ iff $b > 0$

Probability with Martingales: It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because $$\lim aS_n - bn = -\infty \tag{*}$$ but how to prove $(*)$?
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34 views

$\frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx$ - Normal Distribuition

I have read in one of my finance books (Asset Pricing - John H. Cochrane) that there is this identity: \begin{equation} \begin{split} \frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx &...
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82 views

Compound Distribution — Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
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21 views

Check if $Cov(X_1+X_2,X_1-X_2)=0$, i.e. if independent?

"Let $X_1$ and $X_2$ be independent, $N(0,1)$-distributed random variables. Show that $X_1+X_2$ and $X_1-X_2$ are independent." I know that for multivariate normal distributions independence can be ...
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100 views

Variance of a normal distribution for coin toss.

I have difficulties constructing the normal distribution for (20) coin tosses. (Don't ask why, but I never had probability in school.) What is the probability of getting at most 12 heads out of 20 ...
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51 views

how to find the cdf of X in terms of Z when $X=2Z+1$

Consider Z a Normal (Gaussian) random variable with mean 0 and variance 1. It has density $$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} \text{for all x real numbers}$$ We consider $X=2Z+1$. Write ...
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52 views

Probability more than 25% greater?

The random variable X is distributed N(60,64). The random variable Y is distributed N(52,36). Find the probability that a random observation from X is more than 25% greater than a random observation ...
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107 views

Variance of |X-Y| for X and Y ~ N(0,1/2)

I know $X$ and $Y\sim\mathcal{N}(0,\frac12)$, $X$ and $Y$ are independent. I try the following way to solve variance of $g(X,Y)=|X-Y|$ ,which is $V(|X-Y|)$. If $X>Y$,$V(|X-Y|)=V(X-Y)=V(X)+V(Y)=\...
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57 views

Confidence ellipse for a 2D gaussian

For a 1D gaussian, the interval +/- 1SD about the mean will comprise ~68% of the area under the curve. Consider a 2D gaussian with a mean of zero and a diagonal covariance matrix (i.e., it is not ...
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1answer
53 views

To find $\sigma$ of a normal distribution

Given $X \sim \mathcal{N}( n, \sigma^2)$. The question told me $\mathbb{P}(X\lt 3) = \mathbb{P}(X\gt 7)$ So I found $n$ which is $5$. I'm also given $2\mathbb{P}(X\lt 2) = \mathbb{P}(X\lt 8)$. ...
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45 views

Central Limit Theorem and Normal Distribution problem.

Suppose I have a sample of people of size $n$ in which the probability that one smokes is p. I am asked what n should be so that the proportion of smokers in the samples is, in approximation of 0.01, ...
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43 views

Sum and difference of three normally distributed variables

We are given three independent random variables $X, Y, Z$ with normal distribution $\mathcal{N}(1,2)$. Are $U=Z-Y+X$ and $V=X+Y$ independent? I thought I would compute the joint density $f_{UV}$ and ...
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91 views

Integral involving CDF of a normal distribution

Can we evaluate the following integral ? $$\int_0^\infty x e^{-x^2} \Phi(ax+b)\,\mathrm dx$$ Here $\Phi(\cdot)$ is the cumulative probability distribution function of a standard normal random ...
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53 views

Convolutions and the Gaussian distribution

Suppose $X_1$ and $X_2$ are independent random variables each with the standard Gaussian distribution. Compute, using convolutions, the density of the distribution of $X_1 + X_2$ and show $X_1 + X_2 = ...
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176 views

Normal distribution notation

I am wondering... is saying $\mathcal{N}\left(0,\begin{bmatrix} 0.1 & 0.02 \\ 0.02 & 0.3 \end{bmatrix}\right)$ equivalent to $\mathcal{N}\left(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{...
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65 views

How to calculate the Gaussian Integral in specific region?

Firstly, I know that the Gaussian Integral formula, e.g., $\int^{+\infty}_{-\infty}e^{-ax^2}dx=\sqrt{\frac{\pi}{a}}$. But, I am now being encountered a problem when the integral region is not $[-\...
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206 views

z score of normal distribution

Good day, I want to ask about standard normal distribution. What is the highest and lowest value of $z$ score can be? From the table of standard normal, the value $z$ score is only for -3.99 $\leq$ $...
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150 views

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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348 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
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288 views

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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41 views

Probability distritubion of linear function

Given a variable X belongs to gaussian distribution $N(\mu, \sigma)$. How to find the distribution of linear function $y=ax+b$? My answer is that the linear distribtion will belong the $N(a\mu,\sigma)$...
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71 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
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138 views

Bivariate distribution of the sum and product of Gaussian distributed numbers

If $X$ and $Y$ are independent normally distributed random variables $$X,Y\sim\mathcal{N}(0,\sigma^2)$$ How are the sum and product, $X+Y$ and $XY$, co-distributed? You can write the moment ...
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1answer
119 views

A sequence of Gaussian random vectors converges to a Gaussian random vector

Suppose $\left\{X_n : n \in N\right\}$ is a sequence of Gaussian random vectors and $\lim_n X_n = X$, almost surely. If $b := \lim_{n\rightarrow\infty} EX_n$ and $C := \lim_{n\rightarrow\infty} \...
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463 views

Differentiating mahalanobis distance

I would like to differentiate the mahalanobis distance: $$D(\textbf{x}, \boldsymbol \mu, \Sigma) = (\textbf{x}-\boldsymbol \mu)^T\Sigma^{-1}(\textbf{x}-\boldsymbol \mu)$$ where $\textbf{x} = (x_1, .....
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Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal $(\frac{1}{\sqrt{2\pi}})...
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167 views

Sum of Gaussian Variables may not Gaussian

I am currently trying to understand the following three points which we discussed in lectures recently: We say that $X=(X_1,\ldots,X_d)$ is $d$-dimensional multivariate Gaussian distributed if $X\...
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153 views

How to use Chebyshev Inequality

Use Chebyshev Inequality to estimate the probability that in any one day of a business that earns a mean of 100 dollars a day with a standard deviation of 28.87 dollars, that business will make either ...
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138 views

Standard Normal Distribution to Normal Distribution

Given function f() which returns a random value from a standard normal distribution, is it possible to define a function g(mu, sigma) which returns a random value from a normal distribution with a ...
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1answer
45 views

Cumulative Distribution Function of $X = Y*I + Z*(1-I)$

I have a random variable $X = Y*I + Z*(1-I)$ where $Y~N(u, sigma^2)\,,\,\, Z~N(u, sigmac^2)$, and $I~B(1,1-p)$. What I can't seem to figure out is how to get a cumulative distribution function for $...
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578 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
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345 views

Normal Distribution Question

Could someone go over these calculations and tell me where I'm going wrong please. It's to do with normal distribution. The question: In a factory, the packets of sweets produced are supposed to ...
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859 views

Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent)

Would like to know how to approach this question: Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent). Seen the solution for the ...
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209 views

Independent, Normally Distributed R.V.

Working on this: A shot is fired at a circular target. The vertical and the horizontal coordinates of the point of impact (with the origin sitting at the target’s center) are independent and ...
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800 views

Conditional probability distribution with Gaussian noise

If I have a relationship as follows: $$Y = a X + G(0,\sigma^2),\text{ so }y = a X + \text{some Gaussian noise}.$$ The conditional probability distribution of $y$ given $x$, i.e. $P(y|x)$, is equal ...
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156 views

identically distributed question

$X$ and $Y$ are independent normal random variables and have the same moment-generating function and are thus identically distributed. Find the distribution of $Z$ where $Z=aX+bY$. I have the MGF ...
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329 views

Find standard deviation

The assignment is given: In a heat regulated room, we have two temperature limits $T_{\text{min}} = 16$ and $T_{\text{max}} = 24$. If the temperature is between $T_{\text{min}}$ and $T_{\text{max}}...
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126 views

Covariance of bi-variate normal distribution

$ \left( \begin{array}{c} X_1 \\ X_2 \end{array} \right) \sim N\left( \left( \begin{array}{c} 0 \\ 0 \end{array} \right) , \left( \begin{array}{cc} 1 & r \\ r & 1 \end{array} \...
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Characteristics of a Gaussian on a logarithmic scale?

I have data modelled to be Gaussian-distributed, and then displayed on a logarithmic scale and like to know some properties of the displayed data. Effektively, I ignore that a Gaussian $X$ (with ...
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10 views

Chi-Squared Distribution

Let $Z_1, Z_2, Z_3$ be independent standard Normal R.V.'s. Which of the following has a Chi-Square distribution with 1 degree of freedom. $$ \begin{align} A) & & & \frac{Z_1^2, Z_2^2}{2} ...
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33 views

Find exact difference between two values in Normal Distribution

If we have a normal distribution of N(10,2) and we are asked on what is the proportion of values betwen 7 and 8 we can calculate this by: ...
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46 views

Calculate $E(X^{2n})$ where $X$ is normal (0,1)

I need help proving the following: Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that: $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$ I've ...
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22 views

Derivation of a property of standard Wiener processes

I am reading A Standard Wiener Process and am struggling to piece together how they arrived at their conclusion. The major properties of any Wiener Process are: $W(t) = 0$ $W(t) - W(s) \sim N(0, t-...
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17 views

Link between conditional characteristic function and conditional density

Let $X$ and $Y$ be random variables (real-valued). I define $$E[e^{i\theta X}\mid\sigma(Y)] =: g(Y,\theta)$$ Suppose that $g(Y,\theta) = e^{i\theta Y}e^{-\frac{1}{2}\theta^2}$. Can I then say that ...
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Confidence Itervals; $Z_{\alpha}$ & $Z_{\alpha/2}$

I'm confused about what exactly $Z_{\alpha}$ is, does there exist a formula for it in terms of $\alpha$? IF so, is there also one for $Z_{\alpha/2}$?
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34 views

get data to draw a gauss curve

I would like to know how to get some data from a normal distribution to draw its gauss curve. I have the standard deviation, the average and the x, but I don't know how to get some points to draw the ...
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37 views

How to solve $P(X=a) = P(X=b)$

A random variable X is normally distributed with $\mu = 60$ and $\sigma$ = 3. What is the value of 2 numbers a,b so that $P(X=a) = P(X=b)$. The solution is $a = 60$ and $b = 65$. However, I do not ...