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

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Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds$ Basic substitution always gives a $s^{-\frac{1}{2}}~ds$ counterpart which I don't know how to get rid of. ...
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82 views

Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...
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3answers
147 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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75 views

How to calculate the value of $E[X^4], E[X^6],E[X^8] $…?

I learned that when X is a normal random variable , $X$~ $N(0,1)$ , $E[X^2]=1$ $E[X^4]=1.3=3$ $E[X^6]=1.3.5=15$ $E[X^8]=1.3.5.7=105$ For the general case , when variance is s , how do you do ...
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35 views

Normal distribution probability function definition

Up to now, I believed that k-dimensional normal distribution has probability function: $\frac{1}{\sqrt{(2 \pi)^k |\Sigma|}}e^{-\frac{(x-\mu)^T\Sigma^{-1}(x-\mu)}{2}}$ Recently I have read an article ...
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34 views

CLT approximation

Let $X_1,\ldots,X_{735},Y_1,\ldots,Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$ and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Find $P(\sum_{i=1}^{735} X_i ...
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54 views

Probability of random variable in Normal Distribution

I've been talking to my lecturer about choosing random values from a Normal Distribution and he says the following: "Roughly 68% of expected values $ \in (\mu-\sigma,\mu + \sigma)$ does not imply ...
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2answers
56 views

Sampling from a Normal Distribution

If I am sampling randomly from only the -sigma to +sigma interval of a normal distribution and rejecting all other numbers, does it imply that the probability density changes? If so, by what degree? ...
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1answer
36 views

Upper bound for the gaussian measure of an epsilon strip.

I have a question concerning the normal probability distribution: Suppose that $X\sim N(\mu,\sigma)$ is a normal distributed random variable with mean $\mu$ and variance $\sigma$. Let ...
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2answers
299 views

Hypothesis test (when to use z-test or t-test)

I've got 6 questions here. I don't really need the complete answers, I just want to know what test (z or t) should be used and what are the basis for using that test. Here we go: (1) According to the ...
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2answers
72 views

Use z confidence interval to estimate population proportion

Which of the following must be true of a sample in order for it to be appropriate to use a $z$ confidence interval to estimate the population proportion? (A) The sample is a random sample from the ...
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1answer
43 views

How to show that the method to create two correlated random numbers is correct?

I would like to understand how I can show that the method to create two normally distributed random numbers given as an answer to this question is correct. Given independent $X_1$ and $X_2$ normally ...
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2answers
165 views

Distribution of sum of jointly normal random variables with given covariance matrix

Assume that $(X_1, X_2, X_3)$ are jointly normal random variables with the mean vector $(a,b,c)$ and the covariance matrix: $$\left( \begin{array}{ccc} \sigma_1^2 & \alpha & \beta \\ \alpha ...
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1answer
152 views

Is data normally distributed at the 5% significance level?

I have a statistics question I cant wrap my head around: The data sure looks normally distributed as it follows a bell curve and the mean, median, mode could are relatively the same. I just don't ...
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3k views

Distribution of the sum of squared independent normal random variables.

The sum of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then ...
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2answers
244 views

Distribution of angle of two dimensional normal vector

The original subject is: Suppose random variables $X$ and $Y$ are independent and both follow the Normal distribution $N(0,\sigma ^2)$. 1) Prove $U=X^2+Y^2$ and $V = \frac{X}{\sqrt{X^2+Y^2}}$ ...
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236 views

Construction of Gaussian Hilbert spaces

I am reading the very first chapter of "Gaussian Hilbert Spaces" by S. Janson. Definition: A Gaussian Hilbert space is a closed subspace of $L^2(\Omega, \mathcal{F}, P)$ consisting of centered ...
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320 views

Example calculation of estimating GMM parameters using EM

I'm trying to study expectation maximization and I've almost got the idea. What I'm missing is a concrete example. Could someone familiar with the subject give me a concrete example how one would ...
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1answer
217 views

Showing that two Gaussian processes are independent

Say that $Z_t = (X_t, Y_t)$ is a 2-dimensional Gaussian process (by definition, it means that the random vector $(X_{t_1},Y_{t_1},...,X_{t_n},Y_{t_n})$ is a Gaussian random vector for all $t_1 ...
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1answer
120 views

change of unit normally distributed random variable

Assume that $X_{1}$,$X_{2}$,$X_{3}$ are independent continues random variables with $\mathcal{N}(30,12)$, what is the normal distribution of $X_{average}$ (average of $X_{1}$,$X_{2}$,$X_{3}$) ...
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903 views

Probability: Normal Distribution

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability $0.95$. Approximate the probability (by a normal distribution) that at most $10$ of the ...
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173 views

On the total weight of baseballs with normally distributed weights

Assume the weight (in ounces) of a major league baseball is a random variable, a carton contains 144 baseballs. Assume now that the weights of individual baseballs are independent and normally ...
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119 views

Finding the expectation of functions of random variables with a bivariate normal distribution

X and Y have a bivariate normal distribution. I am given that $E[X] = 4$ and $E[Y] = 10$. I am asked to find $E[X^2 - Y^2]$ WITHOUT integration. I know how to solve for this using integration, but ...
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203 views

Calculating integral with standard normal distribution.

I have a problem to solving this, Because I think that for solving this problem, I need to calculate cdf of standard normal distribution and plug Y value and calculate. However, at the bottom I ...
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39 views

Probability , Geometric and Gaussian

So,I'm good at the questions which require the understanding of basic formulaes , but this one my prof said needs me to think (for the first one)'geometrically'=Stumped. Please Help! The second is an ...
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444 views

Calculating an average on normal distribution

Given the fair dice, if the result is $1$ or $2$ the profit is $3$USD, if the result is $6$ you don't win or lose anything, for every other result you lose $2$USD. What is the average profit, that ...
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100 views

Can the characteristic function of a multivariate normal distribution be extended from a neighborhood of the origin?

Let $x$ be a scalar random variable. There is a theorem that states that if $E[\exp(ixs)]= \exp\Big( i{s}\mu - \tfrac{1}{2} {\sigma^2s^2} \Big)$ for some neighborhood around the origin (i.e. ...
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1answer
124 views

The MLE of a $N(\theta, 1)$ distribution

I am trying to find the Maximum Likelihood Estimator of an i.i.d. sample $X_1, \ldots, X_n$ arising from the model $N(\theta, 1)$, where $\theta \in [0,\infty)$. I have done this problem previously ...
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2answers
90 views

Bound of Standard Normal Integral

Consider the Standard Normal Integral given by: $$ I=\int_{-\infty}^{\infty} \frac{1} { \sqrt{2\pi} } e^{ \left( -z^2 /2 \right)} dz $$ In order to prove that it exists we note that the integrand is ...
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341 views

Tail inequalities for multivariate normal distribution

There exists an closed expression for univariate normal CDF, together with simpler upper-bounds under the form, $$ \Pr\big[X > c\big] \leq \frac{1}{2}\exp\Big(\frac{-c^2}{2}\Big)~, $$ $$\text{where ...
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199 views

What proportion are above x of sample size n where X ~ N(0,1) Homework

I have a homework question that I'm not quiet sure of. It follows as so: Consider a random variable $X$ that has a standard normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$. ...
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365 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 ...
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471 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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1answer
179 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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247 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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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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149 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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48 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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102 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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128 views

Finding 'symmetrical range' from mean.

A machine used to make butter where its masses are normally distributed with mean m and standard deviation s.It is found that 5% from these butters are having mass more than 85g where else 10% are of ...
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5k views

Normal distribution with absolute value

I am new to the normal distribution topic. While I have understood and solved various different kind of questions, the normal distribution questions with absolute value, are the ones I have no idea ...
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1answer
148 views

Solve covariance matrix of multivariate gaussian [duplicate]

This is a practical, and basic question. I have a multivariate Gaussian in $M$ dimensions with center $\mu$ (known, lets assume $0$) and some points $p$ where I have the value of $$ \ln(L)= ...
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1k views

Mean and variance of the product of a normally distributed random variable

If a random variable X is normally distributed: $X \sim N(\mu,\sigma^2)$ what is the mean and variance of the random variable $Y = aX + bX^2$, where $a$ and $b$ are constant Given that ${\mathbb E} ...
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2k views

What is the difference between distribution and dispersion?

I need to explain the difference between a distribution (Normal, Chi-square, Poisson, etc.) and Dispersion (as measured by variance, standard deviation) to some students. What is the simplest ...
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332 views

Normal Distribution Probability

At a large publishing company, the mean age of proofreaders is 36.2 years, and the standard deviation is 3.7 years. Assume the variables are normally distributed. a. If a proofreader in the company ...
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169 views

To what extent the statement “Data is normally distributed when mode, mean and median scores are all equal” is correct?

I read that normally distributed data have equal mode, mean and median. However in the following data set, Median and Mean are equal but there is no Mode and the data is "Normally Distributed": $ 1, ...
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666 views

Relation chi-square and student-t distribution

First I want to prove that the sum $Y_1+...+Y_n$ where $Y_i=X_i^2$ and $X$ is standard normally distributed has density $f_n(x)=c_n x^{n/2-1}e^{-x/2}1_{x>0}$ I do not want to derive it, I would ...
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56 views

distribution of Brownian Motion involving integral

What is the distribution of $\int_{t}^{T} W(s)ds$? Given that W(t) is brownian motion. So far, I have the following, $\int_{t}^{T} W(s)ds$ = $(T-t)W(t) + \int_{t}^{T} (T-s)dW(s)$ Also, ...
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42 views

Changing bounds of integrals

If I have: $$\int_{L}^{\infty }e^{\dfrac{-(x-\sigma \sqrt{T})^2}{2}}\,dx$$ let $y = x - \sigma \sqrt{T}$ $$\int_{L - \sigma \sqrt{T}}^{\infty }e^{\dfrac{-y^2}{2}} \, dy$$ Why does the lower bound ...
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probability of sample variance lying between given values

Let $X_1,\ldots,X_n$ be a random sample of size $n = 10$ from a population which is Normally distributed with mean $= 48$ and variance $= 36$. What is the probability that the sample variance of such ...