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

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Normal distribution and option pricing [on hold]

I have a question: A random variable $z$ is normally distributed. Two options have a strike price $k$, then which has a higher value $|z|$ or $z^2$? Thanks for the help!
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1answer
14 views

Why normal approximation to binomial distribution uses np> 5 as a condition

I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success. On most websites it ...
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12 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
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0answers
17 views

Normal distribution exercise [on hold]

In a factory, compacts are filled with a cosmetic powder. We consider the weight of the powder follows a normal distribution $N\sim(\mu, 1.21)$. The value of $\mu$ depends on the setting of the ...
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2answers
48 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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0answers
84 views

PDF of X +Y + X* Y, when X and Y are independent Normal [on hold]

I have $X,Y$ iid Normals $N(0,\sigma^2)$ What is the distribution of $X+Y+YX$? Thnks a lot!
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2answers
41 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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0answers
21 views

Write $\Phi_n(\sqrt{y-1})$ in terms of $\Phi(y)$ and $n$. ($\Phi_n$ CDF of a $\mathcal{N}(0,\frac{1}{n})$)

I'm trying to solve the following problem: Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt $$ Let $F_{Y_n}$ ...
1
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1answer
41 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
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2answers
93 views

Fraction Problem. 3rd grader question got parents thinking

So our nine year old son comes home from 3rd grade and tells us an amazing thing happened in school today. He was playing a math game with his friend and they got the same score two times in a row! ...
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2answers
23 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
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0answers
9 views

how to find the value of n of a normal distribution in R

I have an example that my variable Y is the number of hours (eating say) which follows a N(50,8) distribution. Is there any way to find the number of hours that follows this distribution? Thanks
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0answers
19 views

p.d.f and distribution of multivariate normally distributed variables

Suppose $X\sim N(\mu,V)$ where $\mu = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$ $V = \begin{pmatrix} 3 & 2 & 1 \\ 2& 4 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ a) ...
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0answers
11 views

Confidence Interval Estimate [on hold]

Assume a simple random sample is taken, the conditions for a binomial distribution are satisfied, and the sample proportions can be approximated by a normal distribution. From a sample of $200$ fish ...
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1answer
25 views

Mean and variance: Gaussian is the most conservative assumption

"given only the mean and variance of a distribution, the most conservative assumption that can be made about the distribution is that it is a Gaussian having the given mean and variance" I've read ...
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1answer
18 views

stdev and mean from gaussian fit vs. from classical formula

I have a set of data - measured speed of molecules in water. I made a histogram and fitted it with function $$A\exp\frac{(x-B)^2}{C}$$ calculating mean and standard deviation from values B and C If I ...
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0answers
19 views

Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
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0answers
11 views

Pivotal quantity that is a function of the z-score: find the CI

** Assumptions ** Let: $X$ be a random variable. $\bar{X}_n$ be the sample mean of X; $\mu$ be the expectation value of X; Assume that $\mu$ is not observable; $S_n^2$ be the sample variance of ...
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0answers
15 views

finding probabilty of x=? on z table

I was given a question: what is the probability that $x$ is less than $3 P(X<3)$ This is normal distribution mean $= 2.4 SD = 1.3856$ In the working they said to do the following: $P(X<3) = ...
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1answer
62 views

Statistics: Relationship between process capability and mean

A company produces one-kilogram sugar packets. The specifications on the net content are 1000 ≠ 5 grams. Assuming that the net content follows normal distribution with mean weight as 1005 grams and ...
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0answers
17 views

When $(X, Y)$ are jointly normal given that both $X$ and $Y$ are normal?

We know that $X$, $Y$ are normal does not guarantee $(X, Y)$ is jointly normal. A typical example is: $X=Z$, and $Y=ZU$, where $Z$ standard normal, $P(U=1)=P(U=-1)=1/2$, and $Z, U$ are independent. ...
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0answers
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Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
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1answer
127 views

Finding the distribution function of a random variable using CLT

Let $f_0$ and $f_1$ be two continuous probability density functions with means $\mu_0,\mu_1$ and variances $\sigma_0^2,\sigma_1^2$ on $\mathbb{R}$. Furthermore, let $l(y)=f_1(y)/f_0(y)$ be the ...
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1answer
51 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
11 views

Geometric Sequence with Normal Distribution Problem

Given: The running time (in seconds) of an algorithm on a data set is approximately normally distributed with mean 3 and variance 0.25. a. What is the probability that the running time of a run ...
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0answers
3 views

Finding a Gaussian Distribution to approximate a distribution with non-positive definite covariance matrix

We have got a Gaussian distribution covariance matrix(precision matrix) and the potential information, that is, if g is proportional to exp(-X'KX+h'X). However, K here is not positive definite. So we ...
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1answer
16 views

Moments of maximum of bivariate standard normal

Let $X,Y \sim N(0,0,1,1,\rho): f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}}$, and let $Z=max\{X,Y\}$. I'm looking for the first two moments of $Z$. I know it is ...
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2answers
26 views

normal distribution using Z - finding probability between 2 numbers

I am wanting to find the probability of the following: SD = 20 Mean = 100 P(85 < X < 117) i have found the z values for both: P(X>85) : X-u/o = 85-100/20 Z = -0.75 and found the ...
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1answer
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19 views

Understanding sampling from a normal distribution with zero mean

I'm studying probability. I came a cross "sampling from distributions". Given a probability density function $f_X(x)$, what I understood is that sampling means getting values of $x$ according to the ...
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1answer
14 views

Bivariate normal distribution when $\rho$ is 0

What happens to the bivariate normal distribution when $\rho$ is 0?The bi-variate normal reduces to a simpler distribution, but what is it? and how do you calculate the cdf then? What I have tried: ...
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1answer
37 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
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3answers
71 views

Gaussian integral evaluation

Asked a question to evaluate the Gaussian Integral, $$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx $$ using the the following approximation, $J=\Bbb E[X^2] \sim J_N = 1/N ...
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2answers
28 views

Normal Distribution Problem

The time taken for a computer to connect to a server is normally distributed with a mean value given by 3.3 seconds and a standard deviation of 0.66 seconds. (a) A computer is said to have a fast ...
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0answers
29 views

Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous?

Let $X$ be a standard Gaussian random variable. Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous ? I don't understand the question here. Now $X$ has density ...
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1answer
40 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$
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0answers
16 views

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
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0answers
14 views

Parameters of normal distribution following other distributions

x follows a normal distribution: x~Normal(μ,σ). However, the two parameters of this normal distribution, μ, σ, follow other distribution. Specifically, μ follows normal distribution: ...
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how to use standard normal table?

I don't get it. How do I use the following "type" of table? I can use the ones with negatives just fine, but I don't understand this one: ...
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2answers
51 views

Compute the density of $Y=|X|$

When $X$ has the normal distribution $\mathcal N(\mu,\sigma^2)$ , compute the density of $Y=|X|$ I know ...
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1answer
16 views

using standard normal deviation to calculate mean?

if i have an unknown mean, a standard deviation of 4, and P(X < 8 ) = 0.3085, how do I calculate the mean somehow using the standard normal distribution and it's cummulative function? I know that ...
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0answers
7 views

Calculating standard deviation on CAS software knowing mean and probability?

If have have mean mean A, probability 0.0548 (for X < 20) and standard deviation 5, then I can calculate the mean by doing the following in TI Nspire CAS: nsolve(normcdf(-infinity,20,A,5)=0.0548,A ...
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0answers
20 views

Finding the Probability Limit and Asymptotic Distribution of Xbar-LogYbar

I'm kinda still new to Large Sample Theory and I have already attempted the question. Not sure if I did it right. Based on Kinchin , I know Xbar converges in probability to mu and Ybar converges in ...
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0answers
23 views

Let Xi be iid with EXi = mu and Vxi = sigma^2. Find the asymptotic distribution of Xbar^2

I don't know why I'm having so much trouble with this question. I am supposed to do it in two ways and the first way was using the delta method. And I hope I did it right. However the question ...
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1answer
29 views

The number of coin tosses needed if the proportion of heads is to lie within 0.05 of p with probability at least 0.9?

There's a question I'm not really sure if I did it right or even understand what its trying to say. There is a coin which produces heads with an unknown probability p. How many times should we throw ...
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0answers
21 views

normal distribution variable plus not normal distribution variable

I just started to learn normal distributions and learned that a variable is normal if it has a pdf looks like : [1/(sqrt(2pi)*sigma)]*e^(-(x-mu)^2/(2*sigma^2). Now, i have X~N(0,1) and I need to ...
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1answer
22 views

Probability Distribution of z/x given x

It may seem a simple question for you, but it's driving me crazy. Given the regression model $z = wx + \epsilon$, where $ \epsilon \sim \mathcal{N} (0, (\sigma x)^{2} $, $ z \sim \mathcal{N}(wx, ...
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0answers
5 views

Performance of an optimum estimator for Gaussian random variables used against Non-Gaussian random variables

Consider an optimum estimator for some parameter where the underlying distribution is following a Gaussian distribution with mean 'mu' and standard deviation 'sigma' (denoted by N(mu, sigma)). Let ...
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1answer
28 views

Normal distribution percentile calculation

I'm working out the following problem and there is a part that I am not understanding clearly. The weight distribution of parcels sent is normal with mean value $12$ lbs and standard deviation ...