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

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Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
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19 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 $\|(X_1,Y_1)-(X_2,Y_2)\|$?
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1answer
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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 ...
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2answers
22 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 ...
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1answer
42 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 ...
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1answer
29 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 ...
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1answer
20 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 ...
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2answers
31 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 ...
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1answer
43 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\{ ...
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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 $$ ...
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1answer
34 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: ...
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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 ...
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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 ...
4
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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 ...
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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 ...
2
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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 ...
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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}} ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
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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 ...
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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 ...
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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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0answers
22 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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...
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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} ...
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1answer
105 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 ...
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1answer
29 views

Distribution with density $x^2\operatorname{exp}\{-x^2/2\}$

I came across the probability distribution with density $$ f(x)=\sqrt{\frac{2}{\pi}}\,x^2\,\mathrm{e}^{-\frac{x^2}{2}},\quad x\geqslant 0. $$ Is this distribution known under a certain name? I only ...
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0answers
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The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution

Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ...
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1answer
69 views

Simple calculations of mean, standard deviation, and probability

You are a successful entrepreneur that has developed a new sustainable product that is manufactured through a standard production process. As part of this process, the product goes through quality ...
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2answers
28 views

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve? a) $P(0 < Z < 2.17)$ b) $P(-2.50 < Z < ...
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Normal distribution tables - right or left?

Are the probabilities in normal distribution tables given typically to the right or left of the $Z$ score? One such text I am reading says to the right. However, in my lecturer's exercises, I ...
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0answers
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Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
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1answer
40 views

Generate a uniform distribution from n coin flips

I'm making a computer game and I've reduced the problem into something simple: How can I show the player the number of heads he "tossed" given some number of coins = n? Naive expected value is ...
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15 views

Moments of Multivariate Normal Distributions

I have two questions. Suppose we have two multivariate normal distributions $X \sim N(\mu,\Sigma)$ and $Y\sim N(c\mu,\Sigma)$ where $0<c<1$ is a constant, $\mu$ is a vector and $\Sigma$ is a ...
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Find probability given a binomial and a normal distribution

$X$~$Bin(n,p),Y_n$~$N(μ,\sigma^2)$ Where X is the number of trials taking place, and $Y_n$ is the amount of time the $n$th trial takes (independent of other trials). $Z$ is a new random variable ...