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

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2
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0answers
19 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 ...
0
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0answers
15 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: ...
2
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2answers
54 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
18 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 ...
0
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0answers
58 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 ...
0
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0answers
29 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 ...
1
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1answer
35 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 ...
0
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1answer
24 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
7 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 ...
1
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1answer
45 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 ...
0
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1answer
45 views

derive the mean and variance of $\bar X$ using means of sums rules

I can't find anywhere what the means of sums rules are so i'm confused with this question The random variables $X_1......X_5$ are jointly multivariate normal. Their expectations are $E(x)= \mu_i$ and ...
0
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0answers
24 views

Multivariate Normal Variables

Sorry if this is a bit hard to read. Not entirely sure how to use the MathJax formatting on the site... The random variables $X_1, X_2, X_3, X_4$ and $X_5$ are jointly multivariate normal. Their ...
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0answers
17 views

Residuals normally distributed + no heteroscedasticity = data normally distributed?

If my residuals are normally distributed and no heteroscedasticity was found in the data, can it be assumed that data is also normally distributed when using ordinary least squares regression?
0
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1answer
171 views

Mill's Inequality on normal distribution

Given that $Z \sim N(0,1)$. Prove Mill's Inequality: $$P(|Z|>t) \leq \sqrt{\frac{2}{\pi}}\frac{e^ {\frac{-t^2}{2}}}{t} ~\forall t > 0$$
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1answer
34 views

To calculate $E(Y|X)$ and Var$(Y|X)$.

Suppose $U $ and $V$ are independent and each is distributed as $ N(0,1$). Define $ X$ and $Y$ by $Y=X-1-U$,$ X=2Y-3-V$ . What is $E(Y|X)$ and Var$(Y|X)$ ? Again another questions which I'm unsure ...
0
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0answers
72 views

Find the best predictor and the best linear predictor of $Y^2$ given $X$. Suppose $(X, Y ) \sim N(0, 0, 1, 1, p ).$

Once more, there's another question that I'm clueless on how to start. I should have dropped this course earlier. Suppose $(X, Y ) \sim BN(0, 0, 1, 1, p )$, meaning that $X$ and $Y$ are bivariate ...
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0answers
21 views

Antithetic pair of non-independent normal random variables

Suppose that I have two non-independent normal random variables, X and Y such that $(X,Y)$ has mean 0 and the following variance covariance matrix: \begin{bmatrix} 1 & \rho ...
1
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1answer
25 views

Calculate a probability involving drawings from bivariate normal variables with Xi and Yi i.i.d

There's a question which has been troubling me along with my earlier post. To be honest, I'm not entirely sure on how to proceed. All I know is that if X~N(mu,sigma^2) then P(X < A) = P(Z< ...
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0answers
25 views

Conditional covariance in gaussian graphical models

I have a hypothesis, but I'm not sure if its true. The Wikipedia page states that if the covariance matrix is given by $$\Sigma=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right]$$ ...
1
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1answer
20 views

Calculating number on normal distribution curve

Can someone please let me know if I have this question correct: ...
1
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1answer
33 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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0answers
44 views

Why are the real part and imaginary part of normal distribution function independent?

As I said in title, why are the real part and imaginary part of normal distribution function independent? I need a detail derivation to proof it. Thank you.
1
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1answer
25 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 ...
2
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0answers
39 views

Model selection: geometric mean of the standard deviation.

I have two models that represent a physical process. To determine which model is the best, I make some experiments and compare measured data with data predicted by each of the models. The model with ...
1
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1answer
38 views

Where are they getting this number from?

Here's the question that I'm having a problem with: ...
1
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1answer
23 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 ...
0
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1answer
15 views

Sum of variation for loads

The loads on an electrical network with 10 regions are modelled by considering a base load with mean 20mW and standard deviation 3mW. Variation due to regional load is modelled by considering that ...
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0answers
13 views

Sum of random variables for 2m tape

we use 2 metre tape for distance measurement and that the measurement error for the full tape length has 0 mean and variance 1.5cm^2. Find the mean and the variance if the total distance measured by ...
-1
votes
1answer
106 views

Triangle Distribution, How to find Upper Bound ? if you have median and lower bound

if the lowerbound is 3 and Median is 9, How do I calculate the Upper Bound ? I have been told x is drawn from a symmetric triangle distribution. Im not sure which value to use(I have to sub it into a ...
2
votes
2answers
31 views

Normal Distribution in

I am so confused with this problem: The middle 95% of adults have an IQ between 60 and 140. Assume that IQ for adults is normally distributed. a. What is the average IQ for adults? The standard ...
0
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1answer
58 views

Normal Distribution

Would greatly appreciate any help on this homework question, I will post my answers to parts a) and b) underneath as well but I don't think they are correct.Thanks! a) Take 10 different samples of ...
0
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1answer
33 views

$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$?

$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$? Standardizing, $P(\frac{X - 5}{\sigma} < \frac{-1 - 5}{\sigma}) = 0.1587$ $P(Z < ...
0
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0answers
72 views

Geometric Mean of Uniform random variables convergence

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
1
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0answers
67 views

The characteristic function of a multivariate normal distributed random variable

The characteristic function of a random variable $X$ is defined as $\hat{X}(\theta)=\mathbb{E}(e^{i\theta X})$. If $X$ is a normally distributed random variable with mean $\mu$ and standard deviation ...
0
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1answer
95 views

Obtaining useful information from graph obtained via Monte-Carlo Simulations

I've been running Monte Carlo Simulations on some Matlab code and then plot the graph shown below. I was just wondering what useful information I could collect from this graph? Edit: fit ...
0
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1answer
22 views

Condition on variable to make events independent

where, $$n_1,n_2,...,n_M \sim N\left(0,\frac{N_0}{2}\right) $$ how the condition on n_1 makes the events independent ? what is "n_1=n"
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0answers
8 views

Expectation of log((w^Tx)^2) with respect to a multivariate Gaussian

I am interested in solving (or approximating) the following expectation. $$\int_x \mathcal{N}(x|\mu,\Sigma) \log((w^Tx)^2) dx$$ where $x,w\in R^D$ and $w$ is a constant vector. ...
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0answers
6 views

How Do I Determine Significant Skew?

How do I determine whether a skew is significant or not? I define a significant skew as one greater than 2 x sqrt(6/N) Brown, J. D. (1996). Testing in language programs. Upper Saddle River, NJ: ...
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0answers
51 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
0
votes
1answer
14 views

Confidence interval for a binomially distributed observation with few trials?

If there are few trials and you want to get the confidence interval of a binomially distributed observation, is it still okay to use the normal approximation interval, or is that only accurate for a ...
0
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0answers
12 views

Expectation of a function of multivariate normal cdf

Can someone help me find the following expectation $E_Y(Y*\Phi_k(a+BY|\eta,\Omega))$ where $Y \sim N_n(\mu,\Sigma)$ ? I know that $E_Y(\Phi_k(a+BY|\eta,\Omega))=\Phi_k(a|\eta-B\mu,\Omega+B\Sigma ...
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1answer
41 views

Probability of occurrence after Latin Hypercube sampling and then random sampling

I am using Latin Hypercube sampling to obtain numbers from a Normally Distributed set of data, so that I get a uniform spread of numbers across the Normal Distribution. I then select a number at ...
0
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1answer
19 views

Probability of numbers within a Latin Hypercube

What is the probability of occurrence of numbers in a Latin Hypercube? If I have a 1 dimensional Latin Hypercube of 1000 numbers would the probability of each number just be 1/1000? Essentially, I am ...
0
votes
1answer
30 views

How to draw the Curve for this Normal Distribution

This is not my homework question, I am preparing for the GRE test and stuck on it. Question) The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentile ...
2
votes
2answers
39 views

Distribution of $U=\frac{X}{\| X \|}$ and $R^2 = \| X \|^2$ where $X=(X_1, \dots , X_n)$, $X_1, \dots, X_n \sim$ N(0,1) i.i.d. Independence?

I have the following problem: Let $X=(X_1, \dots , X_n)$, $X_1, \dots, X_n \sim N(0,1)$ i.i.d. What is the distribution of $U=\frac{X}{\| X \|}$ and $R^2 = \| X \|^2$. Are $U$ and $R^2$ independent? ...
2
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1answer
30 views

Add Chi-Squared Distribution to Normal Distribution

Let $z \sim N(\mu,\sigma)$. What is the distribution of $z^2+6z+1$?
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2answers
63 views

Conditional multivariate normal pdf with inequality $f(x_1 | x_2 > a)$

Let $$\begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim\mathcal{N}\left[\begin{pmatrix} 0\\ 0 \end{pmatrix} ,\begin{pmatrix} \sigma_{1}^2 & ...
2
votes
1answer
45 views

Is my understanding of the Central Limit Theorem correct?

Have I got this correct - Say we have a population. We take a random sample of size $n$ from this population. I.e. we form a sample $S$ based on random variables $X_1, X_2, ..., X_n$ taken from this ...
0
votes
1answer
48 views

cumulative standard normal distribution formula

I need to calculate a P-value (for significance checking) out of the Z value, mean(0), standarddeviation(1), normal distrubution being cummulative. Is there a function in PHP that could do that? ...
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1answer
43 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 ...