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

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Posterior of mean given an observation from a bivariate normal with unknown but common mean, and known variance

suppose the sample vector $(x,y)$ is generated from a bivariate normal: $$ \left[\begin{array}{c} x\\ y \end{array}\right]\sim N\left(\left[\begin{array}{c} \theta\\ \theta ...
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20 views

Gaussian (Normal) Distribution parameterization [on hold]

I want to parameterize a Gaussian distribution function so as the majority of observations to be close 80% between -1σ and 1σ (the deep blue part of this) in the result curve instead of 68.2%. Does ...
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1answer
19 views

In a normal distribution, x = 32.8 is where the top 30% of the data starts, and x = −54 has a standardized value of -1.65. [on hold]

(a) What is μ and σ for this distribution? (b) What percent of the data falls between x = −39.2 and x = 22?
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1answer
45 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
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1answer
25 views

$P(X=c)=0$ for normally distributed $X$?

Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?
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1answer
34 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
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10 views

Iso-density locus of Gaussian mixture distribution

I would like to known what is the equation of the iso-density locus of a Gaussian mixture distribution. Is such an iso-density locus a union of ellispoids? Let's say that this Gaussian mixture is in ...
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1answer
27 views

What does rotational invariance mean in statistics?

What does rotational invariance mean in statistics? The property that the normal distribution satisfies for independent normal distributed $X_i$, $\Sigma_i X_i$ is also normal with variance $\Sigma_i ...
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1answer
22 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
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2answers
23 views

Adding and Subtracting Normal Distributions

I think I know how to do this, but I'm not sure. I'm just hoping to check myself here before I do a bunch of work incorrectly. Suppose you have three independent normal distributions: Distribution A: ...
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0answers
19 views

How to do sampling for the following problem. [closed]

There are 200 students with a mathematics exam marks. According to marks students are divided into five categories 0-20,20-40,40-60,... and I want to choose two random sample with 25 for a group. ...
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1answer
37 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
3
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1answer
26 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
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1answer
38 views

When using the Central Limit Theorem, how to scale the mean and variance depending on the number of samples?

So I'm reviewing my notes for the central limit theorem for my final and I'm getting hung up on one detail. The two questions below both utilize the central limit theorem, but they use it in ...
2
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1answer
29 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
3
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1answer
31 views

Conditional probability for two normal distributed variables.

I haven't had to do much with probabilities since university, so please excuse if this is trivial or the question is not well specified. Let $X$ and $Y$ be two independent, normally distributed ...
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1answer
19 views

Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
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1answer
20 views

Is there a rule that can be used to easily approximate the pdf(x) for normal distribution?

Given the Normal Distribution with mean Mu and variance Sigma. With the respect to the rule of 3 Sigma, can one use similar estimations for the value of probability density function within 1, 2, ... ...
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0answers
11 views

I want Find finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions [closed]

Write the finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions. Thanks for help.
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1answer
28 views

Word problem related to normal distribution

Consider the following problem: WINK, Inc. is made up of 450 employees who work a total of 13,500 hours per week. If the number of weekly work hours per person has a normal ...
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1answer
70 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
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1answer
17 views

Expected value of lognormal distribution.

Hi I'm stuck on this question: Recall that X is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if log(X) is normal with mean $\mu$ and variance $\sigma^2$. Suppose X is ...
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0answers
21 views

normal approximation to poisson

Cotton yarn is wound onto bobbins, each of which takes $100$m of yarn. If the thread breaks before $100$m is reached, the bobbin is rejected. In a trial of a new spinning machine, $13$ bobbins out of ...
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2answers
37 views

The product of a normal and Bernoulli variables, independent from each other

Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that: \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} & \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 & ...
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0answers
25 views

Why we can use normal distribution to approximate binomial distribution when n is large enough?

Prove why we can use normal distribution to approximate binomial distribution when n is large enough. Hint: Try to read something on bernoull ...
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0answers
12 views

questions about distribution of multivariate normal

I'm looking at this past exam question, For A) Cbhat~N(CU,C(summation)C') B)I have very faint idea of what to do, I tried finding some theroems about distribution but couldn't find any that ...
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2answers
89 views

Evaluating Integral involving exp

I am stuck at the following integral :- $$ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-x^2-\frac{1}{x^2}\right)\,dx$$ Can anybody give me some hint. and also for this function $$ \int_{- ...
2
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1answer
15 views

Probability of an event occurring more than X% of the time

Cans are a normal random variable with a mean of 7.96 ounces and a standard deviation of 0.22 ounces. Suppose that you draw a random sample of 34 cans. Use normal approximation to find the ...
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1answer
27 views

Transforming a normal distribution to a uniform one

I'm searching for a method that transforms a normal distribution into a normal distribution. I've looked everywhere, but I'm not sure if I just missed something completely obvious, if this actually is ...
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0answers
16 views

How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$

Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard ...
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2answers
32 views

Product of a Continuous and Discrete Distribution

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
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1answer
30 views

Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} ...
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1answer
20 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
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1answer
31 views

Mixed Conditioning - Two Normal Distributions

Let $Z \sim \mathcal{N}(0,1)$ and $Y|Z \sim \mathcal{N}(Z, 1)$. Show that $f_{Z|Y}(z|y)$ is a normal density, and find the parameters of this density. What I have so far: \begin{align*} ...
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0answers
15 views

Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
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1answer
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Identity involving the relation Normal Distribution and Other arbritary Distribution

Let $X$ be a continuous random variable taking value at $\mathbb R$ with distribution $F_\theta$ and density $f_\theta.$ Define a function $\psi_\theta:\mathbb R\mapsto \mathbb R$ by $$ ...
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2answers
26 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0$ + integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.
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Simulate and present normal distribution [migrated]

My task is to compare different methods of simulating normal distribution. For example, I use following code, to generate 2 vectors, each 1000 values (Box-Muller method): ...
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1answer
29 views

Expectation of random varible with normal distribution composed with exponential [duplicate]

I am trying to find $\mathbb{E}(e^{-X})$ where $X$ is a random variable with a general normal distribution. I end up with $$(2\pi \sigma)^{-\frac{1}{2}} \int_{-\infty}^{\infty} ...
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1answer
25 views

How to differentiate the standard normal deviation w.r.t. a parameter inside the upper bound

Given that $$N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{s^2}{2}}\:ds$$ And that $$d=\frac{1}{\sigma\sqrt{\tau}}\ln\left({\frac{S}{e^{-r\tau}K}}\right)+\sigma\sqrt{\tau}$$ How do I take the ...
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2answers
18 views

Is it correct that the normal approximation is just approximation of the normal distribution?

In mathematics statistics. I'm a bit confused by the terminology normal approximation. What is it? Is it just something you say when you approximate, for example the normal distribution?
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1answer
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Normal distribution problem; distribution of height

The problem is: the height of children in age from 3.5 to 4 years is described by normal distribution with parameters $\mu =103$ centimetres and $ \sigma=4.5$ centimetres. What is the percent of ...
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how to prove $\mathop {\lim }\limits_{n \to \infty } {\{\Phi [(1 - \varepsilon )\sqrt {2\log n} ]\}^n}=0$?

$\Phi (x)$ is the distribution function of standard normal distribution. $\varepsilon$ is some positive tiny number that is less than 1. How to prove this beautiful and important limitation: ...
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1answer
19 views

Let $Y_{1},Y_{2},…,Y_{n}$ be a normal distribution where $\mu =2$ and $\sigma = 4$. Find $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$

Let $Y_{1},Y_{2},...,Y_{n}$ be a random sample from a normal distribution where the mean is $2$ and the variance is $4$. How large must $n$ be in order that $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$. ...
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1answer
22 views

problem regarding multiple random variables and normal distribution

I try to solve the following out of an old book on statistics: Cardboard boxes are stacked. The boxes have an average height of 10 cm and the height is normally distributed with a standard ...
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1answer
19 views

If the tubes are shipped in boxes of $1000$, how many wrong-sized tubes per box can doctors expect to find?

The cross-section area of plastic tubing for use in pulmonary resuscitators in normally distributed with $\mu = 12.5mm^{2} $ and $\sigma = 0.2 mm^{2}$ . When the area is less than $12 mm^{2}$ or ...
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0answers
16 views

Explanation of Approximation for Integral Over Gaussian Distribution

I am reading an optics textbook that uses the following integral to evaluate the power squared in the lower tail of the following Gaussian integral. $$\frac{1}{{{\sigma _P} \cdot \sqrt {2 \cdot \pi } ...
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1answer
38 views

In a game, $0.38$ buy hotdogs, how large an order should she place if she wants to have no more that a 20% chance of demand exceeding supply?.

A sell-out crowd of 42,200 is expected at Cleveland's Jacobs Field for next Tuesday's game with the Baltimore Orioles, the last before a long road trip. The ballpark's records from games played either ...
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1answer
34 views

Probability that out of their next 100 free throws, they will make between $75$ and $80$, inclusive in basketball game.

State Tech's basketball team, the Fighting Logarithms, have 70% foul-shooting percentage. (a) Write a formula for the exact probability that out of their next 100 free throws, they will make between ...
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1answer
26 views

68–95–99.7 rule mean normal distribution

if I have data that satisfy 68–95–99.7 rule, does it mean the data is normally distributed? Thanks