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

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83 views

Probability of Cholesterol levels

If the mean serum cholesterol level is 217 and the variance is 750, then what is the probability that a randomly selected person would have: A. Cholesterol value between 150 and 250 B. Greater than ...
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33 views

Frechet differentiability, asymptotic normality

I try to prove the asymptotic normality from the Frechet differentiability. Consider $$T(G)-T(F)=L_{F}(G-F)+o\left(d_{\star}(G,F)\right)$$ and ...
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19 views

Linear Gaussian system, covariance of the normalisation constant

If we have the following multivariate Gaussian distributions: $$p(x) = N(x|\mu_x,\Sigma_x)$$ $$p(y|x) = N(y|Ax + b, \Sigma_y)$$ Now how can you deduce p(y) ? p(y) is called the normalisation ...
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57 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
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0answers
21 views

estimate normal distribution parameters by $n$ largest samples

If I have the $n$ largest out of $m$ values of a sample from independent normal distributed random variables $\mathbb{X}_1,\dots,\mathbb{X}_m\sim\mathcal{N}(\mu,\sigma)$ with unknown parameters ...
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1answer
27 views

Can I conclude the following about bivariate normal RV?

If $(X,Y)$ is bivariate normal with mean $[0, 0]$ and variance-covariance matrix $ \left[ \begin{array}{ccc}1 & \rho \\ \rho & 1 \end{array} \right]$ and $Z=-X$ then is it true that $(Z,Y)$ ...
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27 views

What is the probability the maximum sample value comes from one of two random distributions?

Let $X_1$ and $X_2$ be randomly distributed variables with means $\mu_1$ and $\mu_2$ and standard deviations $\sigma_1$ and $\sigma_2$. Samples of size of $n_1$ and $n_2$ are drawn from each ...
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0answers
48 views

conditional expectation of squared standard normal

Let $A,B$ independent standard normals. What is $E(A^2|A+B)$? Is the following ok? $A,B$ iid and hence $(A^2,A+B),(B^2,A+B)$ iid. Therefore we have $\int_M A^2 dP = \int_M B^2 dP$ for every ...
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1answer
154 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
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1answer
74 views

Why is it so easy to marginalize a multivariate random distribution?

From wikipedia: To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to ...
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20 views

proof of As ~ N(A$\mu$, A$\Sigma$A')

assume that s is a vector of states which is distributed according to a gaussian with mean $\mu$ and variance $\Sigma$. A is the state transition matrix How can I proof that As ~ N(A$\mu$, ...
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60 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
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3answers
250 views

On Pr(X>Y) when X and Y are independent normal [duplicate]

Let X∼N(6,1) and Y∼N(7,1) be two independent normal variables. Find Pr(X>Y). the answer is 0.2389 but I do not know how to do it.I have tried adding them and subtracting but i am still clueless.
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1answer
55 views

Calculating the distribution of the average height - normal distribution

I am not sure how I am supposed to work this question out but I am given that the height of students from college A have a distribution written as: $A$~$N(1.78,0.06^2)$ and the height of students ...
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1answer
61 views

Binomial distribution . Heads and Tails

Consider a coin with P(Heads) = 2/ 3 . We toss this coin 100 times (assume that the tosses are independent). Determine the probability that we get exactly 45 tails out of the 100 tosses. First, ...
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0answers
86 views

Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
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1answer
36 views

Confidence interval and normal distribution

For question (a), is the answer 0.7143? For question (b), is the answer 10.85 and 11.95 ?
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2answers
45 views

Average of two incomes, taken from a normal distribution

In a large corporation, people over age thirty have an annual income whose distribution can be approximated by a normal distribution with mean 60,000 and standard deviation 10,000. Two people are ...
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0answers
72 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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0answers
113 views

Multivariate Distribution & Bayes Rule

Suppose I have that an unknown vector, x, where x is drawn from the following distribution$ \bigl(\begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \bigr)$ ~ $N\bigl(0, \bigl[\begin{matrix} \sigma^2_1 ...
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0answers
353 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
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1answer
63 views

If X is log-normally distributed prove the distribution function in terms of standard normal distribution?

I am not being able to solve part C and part D. Somebody please help! Thanks
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60 views

Conditional multivariate normal distribution

If $X = [X_1,\dots,X_n]$ is follows a multivariate normal distribution $\mathcal{N}(\mu,\Sigma)$, are there any (closed form) results known for the distribution of $[X_1,\dots,X_i \mid l_{i+1} < ...
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2answers
106 views

How big of a sample size is necessary to be sufficiently confident in predictions?

A doctor at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be $90\%$ confident that her estimate is within $2$ ...
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1answer
77 views

Generating 2D random vector from 4D covariance matrix

I have such covariance matrix $C$: ...
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1answer
473 views

MLE of fourth moment of normal distribution

Take $X\sim N(0,\theta)$, and let $\phi = E(X^4)$, the fourth moment. What is its MLE, $\hat{\phi}$, and what is the asymptotic distribution of $\sqrt{n}(\hat{\phi} - \phi) $ as $n\to \infty$? Any ...
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3answers
81 views

$Z$ score probability

I was given a question where I was supposed to find the probability of obtaining $y$ between two scores, however when I input my answer it tells me that I'm wrong, the question is given below along ...
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0answers
158 views

Fourier transform of a complex exponential with quadratic argument

I'm a PhD student who is starting to work right now in the well-established field of ultra-fast optics. The thing is that, in most of the papers I have been reading during the past few days, there is ...
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1answer
67 views

Bigger than and equals rewritten in normal distribution question

So it is correct to say that $P(482\le x \le 510) = P(x \le 510) - P(x < 482)$ where x is a random variable in a normal distribution? Thanks!
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2answers
152 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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0answers
128 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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32 views

Finding the distribution under a new measure

Suppose the the value of an stock is $S_t = S_{t-1}exp(\mu +\sigma Z_t) $ where $Z_t$ are standard normal variables. Find the distribution of ln($S_1/S_0$) under the Q measure given that dQ/dP is ...
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33 views

Confusion related to gaussian

I have this confusion related to gaussian distribution Do we need to have something like $e^{-\frac{x^2}{2}}$ to be called gaussian or $e^{-{x^2}}$ is enough to be called Gaussian. I was reading this ...
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218 views

Almost sure convergence of maximum in a sequence of Gaussian random variables

Let $X_1, X_2,\ldots,X_n$ be an i.i.d. sequence of standard Gaussian variables and $M_n=\max(X_1, X_2,\ldots,X_n)$. I am trying to understand the mechanics of the proof of almost sure convergence ...
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24 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 ...
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48 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 ...
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1answer
1k 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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54 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 ...
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94 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
47 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 ...
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1answer
28 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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146 views

Berry-Esseen Theorem-like result with fourth central moment instead of third absolute moment

Let $X_i$, $i=1,\ldots,n$ be i.i.d. random variables with $E[X_i]=\mu$, $E[(X_i-\mu)^2]=\sigma^2$, and $E[(X_i-\mu)^4]=\kappa$. I am interested in approximating the distribution of ...
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27 views

Confusion related to predictive distribution of gaussian processes

I have this confusion related to the predictive distribution of gaussian process I didn't get how the integration gave that result. What is P(u*|x*,u). Also how come the covariance of the posterior ...
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1answer
142 views

Creating an offset bell curve

This is half programming and half math, but I need the math portion answered as I'm no good at it. I have a list of 10 objects and am randomly selecting and object from that list. I need the ...
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896 views

find probability in normal distribution

i would like to check myself if following my answer is correct: let us consider following problem: Suppose the heights of a population of $3,000$ adult penguins are approximately normally distributed ...
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1answer
51 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
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1k views

calculate probability without table

my question is related to normal distribution,namely as i know in GRE quantity section,there could be question related to normal distribution,but of course we will not have table,o how can we ...
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1answer
59 views

Finding confidence interval for a binomial process using the normal distribution?

See, when I was taught how to find confidence intervals, I always needed the sample variance to use a Student $t$ distribution to form the confidence interval. How does this work in the binomial case ...
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51 views

Calculating Expectation

I want to verify the following equation: $$E[(xe^{aY-\frac{1}{2}a^2}-b)^+]=x\Phi(l_1)-b\Phi(l_2)$$ where $Y\sim \mathcal{N}(0,1)$, $\Phi$ the distribution function of a standard normal distribution, ...
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112 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...