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

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Roots of an equation with normally distributed variable

Consider the following equation: $p\left(1-\int _{\mu}^{x} f(y)dy\right) \left[p\left(1-\int _{\mu}^{x} f(y)dy\right)+(1-p)q \right]-xf(x)p(1-p)q=0$, where $p,q \in [0,1]$, $f(\cdot)$ is the ...
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18 views

Probability two correlated normal distributions smaller than a certain value

Let $N_1$ and $N_2$ be two standard normal distributions with correlation $\rho$. What is the probability that $N_1 < T$ and $N_2 < T$? Thanks.
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43 views

Laplace transform of normal distribution function?

In my notes this was left as an exercise and I am a bit rusty with my calculus. Starting with the definitions: $$\mathcal{L}_X(t) = \mathbb{E}[e^{-tX}] = \int_0^\infty e^{-Xt}f(t)dt \;\;\text{ and ...
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21 views

Hessian for inverse probit link

I'm trying to calculate Hessian and Fisher Information for binomial model using inverse probit link, Suppose likelihood function is $L(\pi)=\prod\limits_{i=1}^n \pi_i^{y_i}(1-\pi_i)^{1-y_i}$ and ...
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1answer
45 views

Expected value of a maximum of two draws compared to expected value of each

I am no mathematician, so I apologise in advance for not explaining myself properly, and for asking something that is probably utterly obvious for most of you. The question has to do with the ...
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13 views

Finding probability given a sample from 36 with a given distribution

Let the sample mean and variance be based on a random sample of size 36 from $N(4, 121)$ distribution. Find $P(0 < X < 8, 40 < S^2 < 160)$. Since they are independent it would be ...
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19 views

Matrix normal distribution vs Multivariate normal distribution

I want to know what are the differences between 'Matrix normal distribution' and 'Multivariate normal distribution'? is the below statement correct? 'Matrix normal distribution' is discrete and ...
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28 views

Finding second moment without standard deviation

Just want to make sure I'm doing this correctly Question: The weight limit of a scale is 5.25 kilograms. W is normally distributed with mean weight 5.15 kilograms and standard deviation $\sigma$. ...
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20 views

Expected value of shifted gaussian distribution

If x is a random gaussian noise, how can I derive the value of $$ E[x(n-a)x(n-b)]$$ $$\text{where } x(n)=\mathcal{N}(\mu,\sigma)$$ Also what if x(n) has zero mean? Thanks
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1answer
14 views

Using normal distribution to create confidence interval

Let $Y~N(\mu, 1)$. Use the fact that $P(\left | Y-\mu \right | < 1.96\sigma) \approx.95$ to construct an interval $(a(Y), b(Y))$, such that the probability $\mu$ is in the interval is approximately ...
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23 views

What is the Edgeworth Expansion of the binomial distribution?

For a standardized binomial distributed random variable $\tilde B_n$ we have $$P(\tilde B_n\le x) = \Phi(x) + \frac {q-p}{6\sqrt{npq}} (1-x^2) \phi(x) + ...
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14 views

Marginalizing multivariate-normal distribution canonical form

Regarding the problem of margenalization of canonical forms of multivariate gaussian distribution it was mentioned in probabilistic graphical models text book that $$\int{C(X,Y;k,h,g)}dY$$ is ...
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47 views

Understanding the Normal Distribution?

If a sample is normal with observations independent and identically distributed: $\mu|\sigma^2 \propto N(\beta \,,\,\sigma^2/\, n_0)$ How can I show that $\mu\,|\,x_1,x_2,....x_n\,,\,\sigma^2 \sim ...
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1answer
146 views

Sum of Two Independent Normal Random Variables (Weird Z?)

So by Property $X + Y \sim \mathcal{N}(1+1.5, 0.1^2 + 0.3^2) = \mathcal{N}(2.5,0.1)$ $$P(1<X+Y<1.3) = P\left(\frac {1.0-2.5}{(0.1)^.5}<Z<\frac {1.3-2.5}{(0.1)^.5}\right)$$ Which is equal ...
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3answers
22 views

finding density of $1/Z$ when $Z$ is a standard random normal variable

If $Z$ is a standard normal r.v., we know that its density is $$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2},$$ where $-\infty \leq z\leq \infty$. I want to find what $f_{1/Z}(z)$ is. I let $Y=1/Z$, so ...
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2answers
25 views

find the value a such that the probability that a tire lasts more than a miles is approximately 0.8

The lifetime of a certain tire has normal distribution with $\mu = 50,000$, and $\sigma = 5,000$. (a) Find the probability that the tire will last between 48,000 and 56,000 miles. Let $L$ = ...
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1answer
18 views

When the covariance matrix $\Sigma$ of the p.d.f. of a m.n.d. is a $1 \times 1$ matrix (a scalar)

I was looking at the Wikipedia article talking about the multivariate normal distribution and specifically I was looking at the section talking about the probability density function of that ...
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1answer
77 views

Software developers with $2$ years experience

Statistics suggest that software developer with 2 years of experience in a town earn an average of 70,000 per year, with a standard deviation of 5,000. To verify this salary level, a random sample of ...
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11 views

Show that this conversion between $\Phi$ and erf(z) holds for all z

I am trying to wrap my head around the connections between the standard Normal distribution and the error function. I could use some help working through the following problem. Show that the ...
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1answer
16 views

Finding conditional distribution in multinormal case

[S]uppose that $X_1$ (sales), $X_2$ (price), $X_3$ (advertisement), and $X_4$ (sales assistants) are normally distributed with: $$ \mu = \begin{pmatrix} 172.7 \\ 104.6 \\ 104.0 \\ 93.8 ...
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1answer
31 views

Applying CLT to Poisson Distribution

If $X$ is a Poisson Random Variable with parameter n, how large need n be so that $\mathbb{P}(|\frac{X}{n}-1)|> 0.01) < 0.1$? Attempt: Noting that X is a sum of n identically distributed ...
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7 views

Linear Combinations of Normal Random Variables

Suppose that $X ∼ N(μ_X,σ_X^2 )$ and $Y ∼ N(μ_Y ,σ_Y^2 )$ are independent random variables. Let $S = aX + bY$ for some $a, b ∈ R$, and use a moment generating argument to show that $S∼N(aμ_X ...
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23 views

Problem on Bivariate normal distribution

Let $X_1$ and $X_2$ have a bivariate normal distribution with parameters $\mu_1 = \mu_2 = 0$ and $\sigma_1 = \sigma_2 = 1$ and $\rho = 1/2$ Find the probability that all the roots of $X_1x^2+ 2X_2x + ...
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17 views

Finding the marginal distributions of a binormal random variable

Let $\overline X$ be a binormal random variable with distribution $N_{\overline X}(\overline m, \Sigma)$ where, $\overline X = \left( \begin{array}{c} x \\ y \end{array} \right)$, $\overline m ...
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21 views

Normal $(\frac{n-1}{n})$-percentile asymptotic to $(2\log n)^{1/2}$?

I am working from Durrett's Probability: Theory and Examples, and I have encountered the following question: Suppose that $X$ is normally distributed, and $b_n$ is defined by ...
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58 views

Derivative of Inverse Mills Ratio (Conditional expectation of normal distrbution is strictly increasing)

I'm trying to show that the derivative of the inverse Mills Ratio is bounded between zero and one. Essentially, for a standard normal distribution, I want to show that ...
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29 views

Normal Distribution with equal probability P(x | y)

Hi I am solving one problem based Bayes' formula. I need to calculate the normal distribution of P(x|y). The following data is given. P(x | y = 0) = N(x1,0,1) and P(x | y = 1) = N(x2,0,16) where N ...
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38 views

Conditional Expectation

Suppose that we have two independent random variables, $x$ and $n$. They are both normally distributed: $x \sim N(\mu_1,\sigma_1^2)$, $n \sim N(\mu_2,\sigma_2^2)$. Let: $$y = x + ln(1 + e^{(n-x)})$$ ...
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13 views

Compute mean of posterior distribution given prior

Is there a formula to compute the mean/expectation of a posterior distribution given the prior? In the ridge regression context, for example. I have $Y=X\theta + \epsilon$ where $\epsilon$ ~ $N(0, ...
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2answers
15 views

Probability and using z-table in Normal Distribution

There's a medication whose target dosage is $\mu$. Due to external errors, the actual dosage follows a normal distribution with mean $\mu$ and standard deviation $\sigma$. I'm supposed to find the ...
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2answers
20 views

Normal Probability Distribution compared to Normal Cumulative Probability Distribution

This is likely a duplicate, but can't find it on MSE. Let's say I have a normally distributed population with $\mu=2.75$ and $\sigma=0.25$. If $x$ is a value in the population of interest, using the ...
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1answer
19 views

Algebraic Simplification for proportionality?

Can someone explain how the author get to the last line from the previous line: Its definitely has to do with the idea you can drop constants when you dealing with proportion but I just don't see ...
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normal distribution and confidence interval

this is the text of my problem "One standard test have been used many years in a class. The test can give a max of 100 points. The sum can be considered as normal distributed. $\mu = 85, \sigma = 25$ ...
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46 views

Wick/Isserlis' Theorem converse

Let $X=(X_1, \ldots X_{2n})$ be a random vector with centered (zero mean) Gaussian distribution. Then, the $2n$-point correlation function: $$E(X_1 X_2 \cdots X_{2n})=\sum_{P \text{ ...
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1answer
91 views

Asymptotic standard normal distribution

I need to solve the following exercise. Assume that $X_\lambda$ is Poisson distributed with mean $\lambda$ . Show that $Y(\lambda) = \frac{X_\lambda - \lambda}{\sqrt{\lambda}}$ is asymptotic ...
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1answer
41 views

Conditional Expectation of $X_2^2$ given $X_1$

Consider two Normally distributed random variables, $X_1 \sim N(0,T_1)$ and $X_2 \sim N(0,T_2)$, such that $X_2-X_1 \sim N(0,T_2-T_1)$. How to calculate $E[X_2^2\mid X_1]$?
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1answer
57 views

Expected value of the absolute value of the difference of two random variables

I have to compute the absolute value of an estimator defined as $T_5=\frac{1}{2}E[|X_1-X_2|]$ in order to state if it is unbiased for $\sigma$, where $X$ is distributed as a $N(0,\sigma^2)$. I am ...
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1answer
13 views

normal distribution integrating over region.

X and Y are investment returns which are normally distributed with X~N(1000,250) Y~N(1000,250) for two consecutive years. The tax rates for year 1 is 30%. The tax rate for year 2 is 20%. If there ...
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1answer
38 views

Use the normal model to approximate the binomial to determine the probability of at least 191 passengers showing up

Because many passengers who make reservations do not show up, airlines often overbook flights (sell more than there are seats). A certain airplane holds 190 passengers. If the airline believes the ...
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1answer
44 views

“Trick” to demonstrate expression is a probability density function for the Gaussian Distribution. [closed]

I was looking into a particular method to demonstrate that the following expression is a probability density function for the Gaussian / Normal distribution. (i.e. that the integral = 1) : $$ f_{X} ...
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2answers
25 views

Is Z in standardized normal distribution always positive?

The question asks me to find the mean, given that: $\sigma$ is $0.8$ when not standardized 96% is over 40 I worked out that $Z = -1.75$ in this case, which then would lead me to $$Z = ...
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1answer
24 views

Non-integrability of the pdf of a squared normal random variable

If $X$ is a normal random variable with mean $0$ and variance $1$, then the pdf of $Y=X^2$ is $f_Y(y)=(2\pi y)^{-\frac{1}{2}}e^{-\frac{y}{2}}$ (for $y\ge0$). But $f_Y(y)$ is not integrable at $0$, ...
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Quantile computation for a Bivariate Normal Distrubtion

Is there a way to compute the following Quantile $x$ since $a ,b$ are known: $$P(T_1 < a , T_2 > x ) = b $$ $T_1$ and $T_2$ have a bivariate normal distribution with known mean and covariance. ...
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0answers
41 views

Ammunition Depot: Monte Carlo Method

I was given the following question from a friend of mine and I can't seem to understand it to well: A squadron of 10 bombers attempts to destroy an ammunition depot. The fighter jet flies in the ...
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1answer
57 views

AWG Noise and RMS Voltage

A question says, a channel is corrupted by Additive White Gaussian Noise with zero mean and RMS voltage 20 nV. The probability that the noise voltage is less than a particular positive value c is 0.9. ...
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12 views

Manipulation of this equation into a Gaussian form

I have been going through a paper (http://www.jting.net/pubs/2007/ting-ICRA2007.pdf) and trying to work out the maths. Ultimately, I came to the following expression $$ \bigg[\frac{1}{2 ...
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1answer
35 views

Normal distribution random variable engine blocks

The diameters of cylinders drilled into an engine block vary slightly, being normally distributed with a mean of 12.500 cm and a standard deviation of 0.002 cm. If the diameter of a given cylinder is ...
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8 views

Finding the distribution of the Sample mean of normal distribution (check my working?)

If $X_1,X_2,...,X_n$ are iid $N(\mu, \sigma^2)$, then how do we find the distribution of the sample mean $\bar X = (1/n) \sum_{i=1} ^n X_i$ ? I tried this: $ \bar X \sim N(\sum_{i=1}^n \mu_i/n, ...
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1answer
38 views

Normal distribution random variable probability

The daily exchange rates for the five-year period 2003 to 2008 between the euro (EUR) and the British pound (GBP) are well-modeled by a normal distribution with mean 1.459 euros (to pounds), and ...
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1answer
20 views

Modeling Gaussian components with standard vs exact functions

I'm studying a paper on modeling DNA histograms. It presents two alternative formulas for modeling Gaussian components: Standard form: $G(x) = ...