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

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1answer
15 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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1answer
25 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
318 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 ...
2
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1answer
70 views
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2answers
43 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
43 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
81 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
134 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
76 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
38 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
68 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
28 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 ...
2
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2answers
58 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
29 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 ...
1
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1answer
71 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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1answer
29 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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1answer
77 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 ...
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1answer
53 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 ...
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1answer
519 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
36 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 ...
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0answers
263 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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1answer
29 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
34 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 $$\Sigma=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right]$$ ...
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1answer
30 views

Calculating number on normal distribution curve

Can someone please let me know if I have this question correct: ...
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1answer
69 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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1answer
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 ...
2
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0answers
73 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 ...
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1answer
42 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
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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1answer
20 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
16 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
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1answer
256 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
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2answers
50 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
87 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 ...
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1answer
40 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 < ...
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0answers
286 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
170 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 ...
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1answer
23 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
74 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
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1answer
23 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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1answer
68 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 ...
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1answer
26 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
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1answer
108 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
55 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
38 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
140 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
66 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
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1answer
147 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? ...
1
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1answer
51 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 ...