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

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1answer
11 views

distribution of distance between two points whose coordinates are normal random variables

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?
0
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1answer
13 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
0
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2answers
18 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
1
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1answer
391 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
0
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0answers
35 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...
0
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1answer
24 views

inequality with gaussian cdf and density involved

in my calculations I've arrived at the following inequality $$ |\frac{4\phi(x)(1-2\Phi(x))}{(1+(1-2\Phi(x))^2)^2}| \leq 0.5 $$ where $\phi$ is Gaussian density, and $\Phi$ Gaussian cdf, which can ...
0
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0answers
26 views

How to fit normal cumulative distribution functions

For a normal distribution $N(\mu,\sigma^2)$, we know its cumulative distribution function is $F(x)=\Phi(\frac{x-\mu}{\sigma})$ where $\Phi(x)$ is $cdf$ for standard normal distribution which means $$ ...
0
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1answer
17 views

Calculate P-Value

In a certain area, regulations require that the chlorine level in wastewater discharges be less than 100 $\mu$/L. In a sample of 85 wastewater specimens, the mean chlorine concentration was 98 ...
0
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2answers
29 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
-1
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1answer
41 views

expectation of a linear combinations of iid standard normal [on hold]

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there Any easy way to calculate $\mathbb{E} \left[ 1_{\displaystyle \left\{ ...
0
votes
1answer
32 views

Calculating probabilities for complex random variables

I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: ...
1
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3answers
786 views

Why normal approximation to binomial distribution uses np> 5 as a condition

I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success. On most websites it ...
0
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0answers
23 views

The X & Y coordinates for points on a bell curve / normal distribution?

In Short: I want to give a formula the X coordinate and get the Y coordinate from matching a bell curve. Is this possible? In Detail: I'm trying to program a market simulation and to get a product's ...
1
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1answer
317 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 ...
2
votes
2answers
47 views

How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
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votes
1answer
30 views

Normal distribution with dice

I'm wondering how to control the normal distribution that comes from summing dice rolls only using different numbers of dice, different combination of types of dice (d4, d6, d8, d10, d12, d20) and ...
4
votes
1answer
29 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
0
votes
1answer
28 views

Sigmoid function that approaches infinity as x approaches infinity.

The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity. The ...
0
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0answers
12 views

Regarding the distribution of pivotal functions not depending on their parameter(s)

I have difficulties understanding the part of pivotal functions not depending on their underlying parameters. Let's take a simple example, if Y is a random sample from an $N(\mu,1)$ distribution and ...
3
votes
1answer
404 views

How to plot standard deviation rings of a bivariate normal distribution?

I'm working on a project right now where I have Gaussian distributions, and I want to create a graphic that represents them. I'm not sure how to generate the ellipse that represents say 1 standard ...
2
votes
1answer
20 views

Expectation of product of two correlated gaussian variables

$\newcommand{\var}{\operatorname{var}}$It seems I can not find the answer anywhere, please point it out how to calculate. Here, I have $X$, $Y$,$G$,$X_D$ and $Y_D$,both are Gaussian variables, and ...
0
votes
1answer
34 views

Variance of a Gaussian Random Variable

Show Variance of a Gaussian random variable $N(\mu,\sigma^2)$ and I know $\mathbb{E}(X)^2 = \mu^2$. So I need $\mathbb{E}(X^2)$ = $\int_{\mathbb{R}} x^2 \frac{1}{\sqrt{2\pi\sigma^2}} ...
2
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0answers
42 views

Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
3
votes
1answer
103 views

Bayesian Updating with 1 Signal but 2 Unknowns

Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 different values {${\alpha_1, \alpha_2}$} such that $\alpha = \alpha_1$ with probability $p_1$ and $\beta$ is ...
1
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1answer
22 views

finding variance of gaussian distribution from mean

The Gaussian random variable $X$ can be used to model the number of customers that enter a market in 1 minute at a given time of the day. The mean number of customers that enter the market in 1 minute ...
1
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3answers
18 views

Finding the probability of loss from standard deviation in normal distribution

I am unsure how to approach the following question. The returns from a project are normally distributed with a mean of \$220,000 and a standard deviation of \$160,000. If the project loses more than ...
0
votes
1answer
16 views

Representation of a non-standard normal variable squared

I have come across a representation of a non-standard normal distributed variable square. It is clear for me that assuming $Z_j \approx N\left ( \theta_j, \frac{\sigma^2}{n} \right )$ we can write ...
2
votes
1answer
472 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
0
votes
1answer
324 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
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0answers
11 views

GLM for normal distribution

$Y_i$~N$\left(\mu_{i,}\sigma^2\right)\space \mu^2=\alpha+\log \left(\beta_0+\beta_1x_i\right)\space \alpha\space is\space unkown$ how is this proved to be a Genralized Linear Model? My assumption ...
0
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1answer
24 views

normal approximation of binomial distribution

a school buys 60% of its light bulbs from supplier A and 40% from supplier B. the light bulbs from both suppliers look identical but light bulbs from supplier A have exponentially distributed ...
1
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2answers
32 views

Gaussian distribution raised to a power

Given that $X$ follows a Gaussian distribution $e^{-x^2/2\sigma^2}$, what distribution is followed by $X^{1/3}$? How does one start to solve this problem? I guess it isn't ...
0
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2answers
71 views

Maximum of two skewed normal distributions

Does there exist a means to approximate the maximum of two skewed normal distributions in terms of another skewed normal distribution? To make it clearer, given 2 skewed normal distributions ...
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0answers
11 views

In a normal distribution curve, why is the probability of Z being greater than 1.64 bigger than Z being greater than 0? [closed]

I've found the probability of a point Z in a normal distribution diagram being greater than 1.64 is 0.505 and the probability of Z being bigger than 0 is 0.5. if you look at it, the probability of Z ...
1
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2answers
65 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, ...
2
votes
1answer
499 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
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0answers
22 views

Inequalities in binomial and normal distrubutions

Example Q Foo is normall distrubuted like $$X\sim N(100,15^2)$$ foo of 110 is required. Does that mean that I should find: $$P(X\gt 109) $$ or $$P(X\gt 110) $$ or $$P(X\ge 110) $$ I feel ...
0
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1answer
17 views

How to Normalize the Sum of Two Gaussians

I have the following function: $I(\theta_i) = I_0 + I_1\exp(\mu(\cos(\theta_i - \theta_s) - 1))$. Suppose I have two implementations of this function, whose parameters match with the exception of ...
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2answers
21 views

Statistics Normal distribution [closed]

The annual rainfall in a town has a normal distribution with standard deviation 5 cm. If the rainfall is over 20 cm for a third of the years, find the mean rainfall.
2
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0answers
20 views

Characteristics function and moments of multivariates

I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf This is a paper by Nengjiu Ju who uses talyor series to express ...
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0answers
14 views

95th percentile of the statistic

Suppose $X_1, X_2, ..., X_6$ and $Y_1, Y_2, ..., Y_6$ are independent, identically distributed normal random variables, each with mean zero and variance $\sigma^2>0$. What is the 95th percentile of ...
0
votes
1answer
19 views

Calculate the Probability of a Normally Distributed Random Sample

Please i would like to understand these problems about probability distributions, I can't find a right solution for this problem. I have a variable X which is the level of glucose in blood and is ...
0
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0answers
53 views

expected value minimum of bivariate normal distribution

Let $X,Y$ be jointly normal with density $f(x,y)=\dfrac{1}{2\pi\sqrt{1-\rho^2}}\exp(-\dfrac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2))$. Let $Z=\min(X,Y)$. Show that $E[Z]=\sqrt{\dfrac{1-\rho}{\pi}}$ and ...
1
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1answer
70 views

The probability that the ratio of two independent standard normal variables is less than $1$

Let the independent random variables $X,Y\sim N(0,1)$. Prove that $P(X/Y < 1) = 3/4. $ Could anyone help me prove this analytically? Thanks. Progress: My first thought was to integrate the joint ...
1
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1answer
16 views

Understanding the normalization of a Gaussian

I have a Gaussian defined as follows: $W(\theta) = j * exp(-0.5 * \theta^2 / \sigma^2)$. I want to set $j$ such that $\frac{1}{360}\int_{-180}^{180} W(\theta)d\theta = 1$. I'm using two values for ...
0
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1answer
20 views

variance of multivariate normal

currently trying to compute the first two moments of the multivariate distribution. Got an extremely helpful answer to show that $\mathbb{E}[x]=\mathbb{m}$, with $x \sim ...
1
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1answer
23 views

Would the joint distribution of Normal Random Variable and the distribution of a X bar from the same sample be bivariate Normal?

I know this question is somewhat redundant... but here goes: My text asserts that the joint distribution of $$X_1=N(\theta, 1)\text{ and } \bar X = N(\theta, \frac 1n)$$ is Bivariate normal with ...
2
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1answer
34 views

Normal Distribution Probability with known mean and variance

I believe I am quite close to solving this, but I would just like to double check some of these answers. Two species have different size toes. Lengths of toes of species X is normal distributed with ...
3
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1answer
23 views

Basically Normal Dist question

I'm a little rusty on my probability and would appreciate any help. I think I have done the bulk of the work already anyway, but my question is: If $X \sim LN(1,2)$ find $P(X>1)$ $X$ being ...
0
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1answer
21 views

multivariate normal moment derivation

I am having trouble deriving the mean for a multivariate normal for $\mathbf{x} \sim \mathbb{N}(\mathbf{m},\Sigma)$: $$ \mathbb{E}[\mathbf{x}]= \int_{R^d} \mathbf{x} ...