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

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1answer
289 views

Probability of a normal distribution; more than, less than confusion.

You are interested in finding how many hours a person is willing to wait for a plane. It is found that the time people are willing to wait has a $μ = 5.2$ and a $σ = 1.1$. What is the probability a ...
1
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1answer
324 views

Given a normal distribution, how do you determine a proportion that is outside of a range?

I am presented with the question: The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a ...
1
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0answers
28 views

Normal Distribution and test hypothesis

I have done 3 experiments. For each one of them, I have repeated the same experiment 100 times. Which gives me three sets of 100 numbers. Experiment 1: for number 30 ---> 100 results Experiment 2: for ...
4
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1answer
2k views

Expected values for normal distribution

So I have a practice question on an example exam, and I am a bit stumped by it: Suppose that $X \sim N(1,2)$. Find: $$ E((X−1)^4) $$ and $$ E(X^4) $$ I am a bit confused as to how to proceed. Now, ...
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1answer
379 views

Normal Distribution Quantiles and Value at Risk

I'm preparing an exam, Quantitative Methods for Financial Markets. My book is not really clear for what concerns the calculation of normal distribution quantiles that have to be used in VaR's formula. ...
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2answers
106 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
42 views

Normal distribution with less Probability [closed]

How do I calculate with Excel's formula to answer the question: What is the probability for a student's point is less than 560 by using excel? I know how to ...
2
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0answers
38 views

Combining two circulating normal distributions

I am working in estimating the impact of location error on location based services. My question is listed below. If the error distribution of location estimation follows in general a normal ...
0
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1answer
123 views

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need?

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need? I may not be using the correct terminology so here's a graph: Based on this, if you ...
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1answer
313 views

Standard deviation of less than one

How would I find the approximate percentage of values within a standard deviation of less than one on the normal model? Chebyshev's rule is only used when the standard deviation is greater than or ...
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1answer
178 views

normal distribution derivation

In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html how do these equal? $$ -k\int (x-\mu) dx = -\frac{k}{2} (x-\mu)^2$$ Isn't this the case? $$ -k\int (x-\mu) dx ...
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2answers
99 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
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2answers
58 views

$X$ is half normal and $S ∼ U{(−1, +1)}$. How $Z = SX ∼ N(0, 1)$?

If we chop a standard normal distribution in half and use only the positive side (scaled up by a factor of $2$ to maintain a proper density), then we get the so-called ‘half normal’ density: ...
3
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2answers
574 views

How to approximate the integral of the standard normal distribution.

So I have this eqn. $$ f(x)= \frac {e^ \frac{-x^2}{2}} {\sqrt{2\pi}} $$ I need to find: $$ \int\limits_{-1}^1 f(x)dx $$ So I want to use this series to integrate. I know that: $$ e^x = ...
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2answers
1k 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 ...
0
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1answer
66 views

Approximating the optimal value of a function involving a Gaussian integral

Consider the following function $$ f(\lambda) = \alpha (1+\lambda^2) + (1-\alpha)2\int_\lambda^\infty (x-\lambda)^2 \phi(x) dx $$ where $\alpha \in (0,1)$ and $\phi$ is the standard normal probability ...
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0answers
26 views

Determining Range

$X_i\sim^{iid}N(0,1);\quad i=1,2$ so $x_i$ ranges from $-\infty$ to $\infty$. Now $Y=X_1^2+X_2^2$ so $y$ ranges from $0$ to $\infty$. But how $Z=X_2$ is ranges from ($-\sqrt y$) to $\sqrt y$ ?
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1answer
52 views

Suitability of skew normal for rating task and calculation

in an experiment, I ask participants to rate qualities on a continuous scale. I expect the results to be normal distributed and I am confident that assuming a normal works fairly well for most values. ...
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0answers
54 views

How to learn mixture Gaussian with inequality constraint of component variances

Let $f_1(x)$,…,$f_n(x)$ be Gaussian density functions with different parameters, $\mu_i$ and $\sigma_i$ are the parameters (mean and variance) of the Gaussian component i, and $w_1,\ldots,w_n$ be real ...
2
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1answer
98 views

Given a covarince matrix, generate a Gaussian random variable

Given a $M \times  M$ desired covariance, $R$, and a desired number of sample vectors, $N$ calculate a $N \times M$ Gaussian random vector, $X$. Not really sure what to do here. You can calculate ...
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2answers
937 views

How do you compute this normal distribution?

The question is: Given that X is normally distributed with mean 100 and standard deviation 9, compute the following for n = 16. (a) Mean (Round your answer to the nearest integer.) and ...
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1answer
205 views

convolve probit function with gaussian [duplicate]

I want to prove the following, however, not sure where to start. $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Where $\Phi(\cdot)$ is the probit function, ...
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1answer
2k views

The difference between unbiased/biased estimator variance.

The biased MLE of Normal distribution is: $\hat{\sigma }_{MLE} = \frac{1}{N}\sum_{N}^{i=1}\left({x}_{i} - \hat{\mu }\right)^{2}$ And unbiased is: $\hat{\sigma }_{unbiased} = ...
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1answer
2k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
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1answer
425 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
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2answers
494 views

Characteristic Function of Inverse Gaussian Distribution

The pdf of Inverse Gaussian distribution, IG$(\mu,\lambda)$, is : $$p_X(x)=\sqrt\frac{\lambda}{2\pi x^3}\exp\left[\frac{-\lambda}{2\mu^2x}(x-\mu)^2\right];\quad x>0,\lambda,\mu>0$$ I have to ...
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1answer
97 views

quotient Groups of different normal subgroups. [closed]

Let G have two normal sub groups N and M,|N|=|M|(so that |G/N|=|G/M|).Now consider their quotient groups G/N and G/M.Is it possible that for each g*n(g belongs to G and n belongs to N),there exists ...
0
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2answers
121 views

Normal distribution probability problem.

There are lots of salmon in a pond and their length (in centimeters) obeys normal distribution $N(70, 5.4^2)$. You and your friend go fishing and decide to continue fishing until both of you catch at ...
3
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1answer
109 views

Z scoring and normal distribution

I tackled a question that asked Given the mean and std dev for rainfall in a city, ...
3
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1answer
48 views

Independent representation of correlated $N(0,1)$ variables

Assume that $X_1$ and $X_2$ are correlated $N(0,1)$ variables. Now we can write \begin{align*} (X_1,X_2)^{T}=(\tilde{X_1},\gamma \tilde{X_1}+\sqrt{1-\gamma^2}\tilde{X_2})^{T} \end{align*}where ...
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0answers
61 views

$\frac{\partial}{\partial\theta}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$

Ok, I am working on a problem that consists of the following: I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ...
0
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1answer
265 views

Ratio of dependent chi squared random variables

Suppose that $X=v'A_1v$ and $Y=v'A_2v$, where $A_i$ are symmetric matrices and $v$ a multivariate normal vector with covariance $V$, are chi squared distributed each with its own degrees of freedom. ...
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2answers
1k views

joint probability of two Gaussian

I was studying factor analysis model using a lecture note by Prof. Andrew Ng (http://cs229.stanford.edu/notes/cs229-notes9.pdf). It says $z \sim N(0,I) \\ \epsilon \sim N(0, \psi) \\ x = \mu + ...
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0answers
96 views

Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
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1answer
64 views

Proving MLE for normal distribution

I need to prove that using maximum likelihood estimation on both parameters of normal distribution indeed maximises likelihood function. So, the log-likelihood function for parameters $\sigma$ and ...
3
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2answers
991 views

Sums of Products of Two Normal Variables

Suppose that $X_1 ,\ldots,X_n,Y_1,\ldots,Y_n$ are all independent normal random variables with different means and variances. What is the PDF of the following random variable? ...
0
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1answer
121 views

integrate moments normal distribution between finite limits

Can somebody help me to evaluate the following integral: $$\frac{1}{\sqrt{2\pi}\sigma}\int_a^b x^2 \exp\left(\frac{-x^2}{2\sigma^2}\right)\mathrm dx$$ Answer involving cumulative normal (erf) would ...
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1answer
129 views

Correlation of sums of correlated variables

I'm trying to work out an expression for a correlation of the weighted sums of two r.v.'s with a third r.v. To be precise, I have a trivariate normal distribution: $$\{X,Y,Z\}\approx ...
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1answer
2k views

How do I know if a sufficient statistic is also complete?

For example, for an i.i.d. sample of random variables $X_i$ distributed according to a normal distribution, I found a sufficient statistic—the sample mean. How do I know if this is also complete? ...
2
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1answer
122 views

Gaussian function

I want to scale the Gaussian function $\exp(-x^2)$ to the unit disc. In particular, I wish to represent $\int_0^\infty \exp(-x^2) dx$ as $\int_0^1 g(x) dx$, where $g$ should be the rescaled Gaussian ...
0
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1answer
57 views

probability in normal density function

Q: let X be a continuous random variable with NORMAL DENSITY $$f(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{−(x−\mu)^2/ 2\sigma^2}$$ We know that $\mu = 70$ and $\sigma = 2$. Find $P(68 \leq X ...
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3answers
4k views

Derivation of the density function of student t-distribution from this big integral.

My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am ...
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1answer
46 views

Question on sum of normal variable

I have a small doubt. If X and Y are standard normal variables, is $ Z=(X+Y)/\sqrt { 2 } $ a standard normal variable ? If I am correct, $X+Y$ follows $N(0, 2)$. So, Z must follow $N(0, 2 / \sqrt { ...
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2answers
2k views

Expected value for maximum of n normal random variable

Let $X_1...X_n\sim N(\mu,\sigma)$ be normal random variables. Find the expected value of $\max_i(X_i)$ and $\min_i(X_i)$. The sad truth is I don't have any good idea how to start and I'll be ...
2
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1answer
170 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
59 views

Get random position on surface

I'd like to get a random position on a surface of an object, and also follow it's normals. Example, let's say I have a sphere, I can get all the face, normal and vertex positions and well as their ...
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1answer
32 views

Normal distributions with errors

I'm able to do the following problem: In a road, the speed limit is $80$ km/h. The car speeds follows normal distribution and has average $70$ km/h and standard deviation $6$ km/h. How many percent ...
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1answer
28 views

Calculate failure of component by central limit theorm

A component in a device fails one time per 24 hours (on average). How many spare parts should be in order to verify that the probability they will be enough for one week is 95%? Use central limit ...
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1answer
99 views

distribution of maximum of $n$ Pearson correlations

$\mathbf{x}=[x_1,x_2,...,x_m]^{\top}$ is a vector of length $m$ and $\mathbf{y_1}, \mathbf{y_2}, ..., \mathbf{y_n}$ are similarly $n$ vectors of length $m$. If the elements of $\mathbf{x}$ and ...
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1answer
215 views

Getting a Hermite polynomial expansion of Gaussian with given variance.

I am trying to find an expansion of centered Gaussian - $\frac{1}{\sqrt{2\pi}\sigma}\exp({-\frac{x^2}{2\sigma^2})}$ in terms of Hermite polynomials. Namely to calculate ...