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

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2answers
43 views

Can You Give Me Help On Expected Value Please?

The Question: Let $X_{1}, X_{2}, ..., X_{9} $ be a random sample of size 9 from a normal distribution $N(2,4)$. Let $Y_{1}, Y_{2} , Y_{3}, Y_{4}$ be an independent random sample from a normal ...
0
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1answer
36 views

Is there a lower bound to the standard deviation of a Gaussian (Normal) distribution?

A Gaussian or Normal distribution is defined by the probability density function $$ f(x \; | \; \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi} } \; e^{ -\frac{(x-\mu)^2}{2\sigma^2} } $$ with $\sigma$ as ...
0
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0answers
21 views

If $X\sim N(\mu_x,\sigma_x^2)$ and $Y\sim N(\mu_y,\sigma_y^2)$ what is $\text{Var}{(X/Y)}$

If $X\sim N(\mu_x,\sigma_x^2)$ and $Y\sim N(\mu_y,\sigma_y^2)$ what is $$\text{Var}{(X/Y)}$$ Such that $X$ and $Y$ are independent.
1
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0answers
25 views

Generating random numbers of bell curve distribution

I want to generate random numbers that fit a bell curve distribution. Basicly, I need random numbers from 0 to 1, but I wish to have a high likelihood of it being close to 0.5, but not guaranteed, ...
2
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1answer
19 views

Stats: Confidence Interval and Upper Limit

A random sample of n = 18 E-glass fibre test specimens of a certain type yielded a sample average interfacial heard yield of 40 and a standard deviation of 4. Assume the interfacial shear yield ...
1
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0answers
19 views

Find likely maximum distance from center of gaussian sphere in high dimensions

I am testing a clustering algorithm in high dimensions. I want to see how it behaves as I allow the clusters to get closer and closer, but it must work perfectly for "well separated" clusters. I need ...
0
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0answers
7 views

Mahalanobis distance for statistic moments skewness and kurtosis

I'm experimenting(playing around ;-)) with a set of points that all have the same weight. I compute the gaussian-distribution values: mean and variance from the set of points. Using the ...
0
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2answers
36 views

Expressing the probability of a joint cumulative normal distribution in terms of $\mu$ and $\sigma^2$

Consider two random variables, $x$ and $y$, which are independent draws from the same normal probability distribution: $x,y \sim N(\mu, \sigma^2)$. I would like to calculate the probability that $x$ ...
0
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0answers
21 views

PDF of sum of squares of two normal random variables

I have two normal random variables $X \sim N(\mu_{x},\sigma_{x})$ and $Y \sim N(\mu_{y},\sigma_{y})$. My problem is to find the PDF of $Z = X^{2} + Y^{2}$. I tried in this way: Let $\mu_{x} = 0$ ...
0
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1answer
22 views

Hypothesis Testing Using Critical Values

If my data $X_i\sim N(\mu,σ^2)$ is iid with a sample $n=35$ If my 99% confidence interval is $[.56,1.86]$ where $P(t>2.7238)=.005$ for a $t(34)$ random variable $t$ How would I derive $\bar{X}$ ...
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1answer
39 views

Find the $E[X^3]$ of the normal distribution

Find the $E[X^3]$ of the normal distribution with mean μ and variance $σ^2$ (in terms of $μ$ and $σ$). So far, I have that it is the integral of $x^3$ multiplied with the pdf of the normal ...
0
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1answer
25 views

Which is the distribution of this data set

We have a test with possible scores from 0 to 100 and a sample of 20 subjects which have the score: 87,53,35,90,78,45,65,87,76,57,86,99,67,98,86,79,90,88,86,95. mean=77.35; standard deviation=17.55. ...
0
votes
2answers
27 views

Calculate probability of normal distribution

Question: Suppose that a random sample of 16 observations is drawn from a normal distribution with mean μ and standard deviation 12; and that independently another random sample of 25 observations is ...
1
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1answer
41 views

Expectation of normal distributions

Let $X$ have a normal distribution with mean $µ$ and variance $σ^2$. Find $E[X^3]$ (in terms of $µ$ and $σ^2$). the pdf of this function is $$\frac{1}{σ\sqrt{2\pi}} e^{\frac{-(x-µ)^2}{2σ^2}}$$ ...
1
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1answer
17 views

Yearly demand distribution with monthly demand given

Currently working on this problem. Monthly demand of a product has been observed to follow a normal distribution with mean of $50$ pieces and standard deviation of $5$ pieces. Assume each month is ...
0
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1answer
32 views

Steps to understand that $ \sigma \int_t^T e^{\kappa(s-T)} dW_s $ is distributed $ \sigma \sqrt{\frac{1-e^{-2\kappa (T-t)}}{2 \kappa}} N_{0,1}$

What are the steps to see that $ \sigma \int_t^T e^{\kappa(s-T)} dW_s $ is distributed $ \sigma \sqrt{\frac{1-e^{-2\kappa (T-t)}}{2 \kappa}} N_{0,1}$? My question stems from the euler scheme for $ ...
2
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1answer
18 views

Variance of a time dependant gaussian

I'm trying to find the variance of the following: $$ \int_{0}^{t} N\Bigl(0,\sigma^2e^{-C(t-\tau)}\sin^2\bigl(B(t-\tau)\bigr)\Bigr)d\tau $$ where $N$ is a Gaussian distribution with zero mean and ...
1
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0answers
24 views

Probability distribution of $g^h f f^h g$

We define an $k \times k$ complex matrix $M=[V \, \mathbf{0}]$, where matrix $V$ is $k \times (k-l)$ dimensional and is unitary, and $\mathbf{0}$ is the $k \times l$ zero matrix. Let vector $f$ be a ...
1
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4answers
46 views

Find $\mathbb{E}[W_t^3]$ and $\mathbb{E}[W_t^4]$

I am very stuck on this past paper question. $W_t$ is a brownian motion and find $\mathbb{E}[W_t^3]$ and $\mathbb{E}[W_t^4]$ I thought, since $W_t$ is normally distributed with density function ...
0
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1answer
37 views

Limiting distribution of a sequence of random variables

$X_1$,...$X_n$,... are iid random variables with mean being $1$ and variance being $1$. Let $S_n=\sum_{i=1}^{n}X_i$. Let $\Phi(\cdot)$ be the cdf of standard normal distribution. What is the limiting ...
1
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1answer
25 views

Competing Probability

Suppose I have two cumulative probability distributions: $P(x)$ and $P(y)$. How do I combine these two distributions to find the combined probability that one random variable is lower than the other. ...
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0answers
4 views

Confidence region for both mean and variance of normal distribution

Given a $N\left(\mu,\sigma^2\right)$ population, how would one find a confidence region for $\left(\mu,\sigma^2\right)$? Since $\bar{X}$ and $S^2$ are independent, can I conclude that the confidence ...
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2answers
24 views

which phenomena follow normal distribution?

In nature so many phenomena following normal distribution such as human length, grass length etc. My question is what what natural phenomena must follow normal distribution? Is there any criterion ...
0
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0answers
12 views

Does it make sense that the distribution of average household income by postal codes is LogNormal?

I have average household income for 69787 Postal Codes of a Province in Canada. Does it make sense to fit a log normal to this data? I also have data for the population in each postal code, is there ...
0
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0answers
14 views

Question about checking normal distribution and analysing confidence intervals

I've got another question about normal distribution and its confidence intervals interpretation. Your explanation will respectively help me to better prepare for my examination. One of the social ...
0
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1answer
12 views

Estimating the measure of set and measure of shifted set

Could you please help me solve the following problem: Is it true that for any $\varepsilon>0$ there exist a $\delta=\delta(\varepsilon)>0$ (depends on $\varepsilon$) such that $$ ...
0
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0answers
16 views

Unbiased estimation of parameter with singular matrix.

Given sample $N_p(A\theta,Q)$, where $\theta, Q$ - unknown. A - known $q*p$ matrix, $rankA = q, q<p$. The question is: how can I find unbiased estimation $\hat\theta$ of $\theta$? It seems easy ...
1
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1answer
23 views

Two questions about normal distribution and its calculation

dear community members, I got this question in my handbook for exam preparation. I was trying to calculate the output but I'm stuck with figuring out the correct value. The question is: According to ...
1
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0answers
23 views

Converting log-scaled volume density to number fraction

I have a log scaled volume density distribution, $q_{3,log}$ from which I want to get number fraction, $\Delta Q_0$ with normal scale. So to transform $q_{3,log}$ to $\Delta Q_3$ the used relation is ...
4
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2answers
106 views

Show $X_1$ and $X_2$ have a common Gaussian distribution

Anyone has any idea about the following question? Let $\Bbb E(X_1^2)$ and $\Bbb E(X_2^2)$ be finite. Show that if $X_1$ and $X_2$ are independent and likewise $X_1+X_2$ and $X_1-X_2$, then both ...
0
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2answers
24 views

Show that $Y\sim\Gamma(\frac{1}{2},\frac{1}{2})$

let $X\sim N(0,1)$ be a standard normally distributed stochastic variable and let $Y=X^2$. Show that $Y\sim\Gamma(\frac{1}{2},\frac{1}{2})$, ie. $Y\sim\chi^2(1)$.
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0answers
24 views

Deriving MLE for covariance matrix using Robbins-Monro

I'm having some trouble completing exercise 2.37 in Bishop's Pattern Recognition and Machine Learning text. I'm not reading this text as part of a course, so this is not a homework question. Here's a ...
0
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0answers
11 views

distribution function of exponential of square of normal distribution

I have a question about distribution functions. Assume that $X \sim N(\mu, \sigma^2)$. I want to compute the mean and variance of $Y = \exp(-\frac{X^2}{2h^2})$ analytically. What is the solution?
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0answers
9 views

Mean value given error is distributed ~ N( 0 , .1 )

The true weight of an object is w It is weighed two different times X1 and X2. Then X1 = w + E1 and X2 = w + E2 Where E are the two measurement errors. Suppose the error is iid with a mean of 0 ...
0
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2answers
49 views

Hint for integrating exp(x-x^2)

The function $e^{x-x^2}$ is zero if $x \to \infty$ or $x \to -\infty$ it looks like a normal-distribution-curve with the max. value at $x=0.5$. Has somebody a hint for integrating it from $-\infty$ ...
1
vote
2answers
42 views

$X$ follows normal distribution $\mathcal{N}(0,1)$

$X$ follows normal distribution $\mathcal{N}(0,1)$. Find distribution of $X^2$. My question is why we are allowed to calculate this as follows: \begin{align*}P\left(-\sqrt t \le X\le \sqrt t\right) ...
0
votes
1answer
14 views

Normalisation of a range of numbers to another range

I want to normalise a set of range of values having 0 Min and a Max that is known but can vary; say 22000 and would like to normalise these values from 0 to 300 and also from 0 to 20. I found a ...
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votes
3answers
42 views

How to solve following normal distribution problem? [duplicate]

If $$X \sim \mathscr N(\mu_1,\sigma_1^2)$$ $$Y \sim \mathscr N(\mu_2,\sigma_2^2)$$ $X$ and $Y$ are independent, how to show that $$X+Y\sim \mathscr N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$$ Hint: ...
1
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2answers
46 views

how to find the cdf of X in terms of Z when $X=2Z+1$

Consider Z a Normal (Gaussian) random variable with mean 0 and variance 1. It has density $$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} \text{for all x real numbers}$$ We consider $X=2Z+1$. Write ...
0
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0answers
7 views

Covariance Matrix Update

I am building a Multivariate (in my case 3-dimensional) Gaussian Distribution from a set of 3-dimensional data. I want to update my model with new incoming data. Updating the means of all the 3 ...
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2answers
61 views

distribution of $Y= \exp(-X^2)$ [closed]

I have a question about distribution functions. Assume that $X \sim N(\mu, \sigma^2)$. I want to compute the mean and variance of $Y = \exp(-\frac{X^2}{2h^2})$ analytically. What is the solution?
0
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2answers
30 views

Use standard error of mean or population distribution?

Question: Marks obtained by certain number of students are assumed to be normally distributed with mean 65 and variance 25. If three students are taken at random, what is the probability that ...
0
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1answer
45 views

Normal Distribution and Percentile Word Problem

The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. Students’ scores on the quantitative portion of the GRE follow a normal distribution with mean ...
0
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2answers
48 views

Standard normal distribution find $\mathbb{E}X(X+1)$ and $\mathbb{E}e^{3X^2/8}$

Random variable X has standard normal distribution. Find $\mathbb{E}X(X+1)$ and $\mathbb{E}e^{3X^2/8}$. My attempt is to solve: $\int_{-\infty}^{\infty}x(x-1) \frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$ ...
0
votes
1answer
21 views

Variance of Normal Distribution

Consider $X_1$~normal(4,3) , $X_2$~normal(5,7), and $X_3$~normal(6,4), where $X_1$, $X_2$, and $X_3$ are independent. Obtain variance of $W = 2X_1 - X_2 + 3X_3 + 3.$ Attempt : Since we are ...
3
votes
1answer
41 views

Estimate the probability $P(X > C\frac{(n-1)^p}{\sqrt{n}})$ for $X\sim N(0,1), C>0, p>1/2$

Let $(B_t)_{t\geq 0}$ be brownian motion, let $p>1/2$. I want to show that $$\lim_{t\to\infty} \frac{B_t}{t^p} \to 0 \quad a.s.$$ Atm I'm trying to show that $$\limsup_{t\to \infty} ...
2
votes
4answers
41 views

Mean of a portion of a normal distribution?

How do I calculate the mean of a portion of a normal distribution. In other words, say I have a normal distribution of the heights of adult males. The mean is 70" and the standard deviation is 4". ...
0
votes
1answer
32 views

Show that $T$ is not a sufficient statistic

Let $X_1\: , X_2$ a random sample for $N(\theta ,1).\:$ Show that $\:T=X_1 + 2X_2$ is not a sufficient statistic. I've tried to prove it by contradiction: I assumed that $T$ is sufficient. That ...
0
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0answers
4 views

what does mean “the distribution $F$ has heavier tails than normale law”?

What does mean geometrically "the distribution $F$ has heavier tails than normale law" ? Same question with "lighter tails". I can't see on a graph and I didn't found example on the internet. For ...
3
votes
1answer
78 views

Calculating an Exponential Integral

Calculate $\int_0^{\infty} \frac{x^{2N+1}}{a+x^{-b}} e^{-c x^2} dx$ where $a,c > 0$ and $b>1$. The best I could do about this integral is to find an upperbound for it: $\int_0^{\infty} ...