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

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0answers
17 views

Standard normal distribution hazard rate

Is the hazard rate of the standard normal distribution convex? Can you give a reference?
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1answer
34 views

Statistics - normal distribution problem

Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. If $P(X > 3) = 0.8413$ find $P((X+Y)/2 > 3)$. The result must be exact number, so normal distribution parameters are ...
0
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2answers
25 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
2
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1answer
35 views

Show that $W$ is a Gaussian process

I have the following problem: I want to prove that the vector $(W(1_{[t_0,t_1]}),...,W(1_{[t_{n-1},t_n]}))$ is normally distributed with mean $0$ and covariance matrix ...
1
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1answer
18 views

Cholesky decomposition and variance

Well, I've been reading about simulating correlated data and I've come across Cholesky decomposition. Everything seemed clear until I found a couple of posts on this site and Cross-Validated that ...
1
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0answers
17 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
2
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3answers
98 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
5
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1answer
64 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where ...
3
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1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
0
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1answer
14 views

Probability of return with 7% error

I have a problem understanding the answer of the following problem: A recent audit by the IRS of the returns she prepared indicated that an error was made on 7% of the returns she prepared last ...
1
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4answers
50 views

The normal distribution is a common model of randomness

Can someone please comment/elaborate on the statement: "The normal distribution is a common model of randomness." I would like to understand it more deeply. Source: Perhaps someone can point me ...
4
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1answer
29 views

Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
1
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1answer
36 views

Statistics- Finding Probability

A local lawn service has determined the average time it takes to mow an average residential yard is thirty-five minutes. If mowing times are independent and constant, what is the probability it will ...
5
votes
1answer
66 views

Checking the Lindeberg condition (central limit theorem)

Problem. Let $W_1, W_2,...$ be independent and identically distributed random variables such that $E(W_1)=0$ and $\sigma^2 := V(W_1) \in (0,\infty)$. Let $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j ...
2
votes
2answers
29 views

Gaussian function constant

Why are 1D gaussians defined as $$F(x;\sigma^2) = e^{\frac{-x^2}{2\sigma^2}}$$ for a probability function (after computing the gaussian integral): $$p_F(x;\sigma^2) = ...
0
votes
1answer
28 views

Probability of agreeing to do some work depending on the payment

I am looking for several options of modeling the probability of people agreeing to do some work depending on the price/payment. The payment can only range between $p_1$ and $p_2$, $(p_1 < p_2)$. I ...
1
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1answer
31 views

Question regarding the derivation of the distribution of $(n-1)S^2/(\sigma^2)$

I will quote from my statistics manual: "Consider a random sample $x_1,x_2...x_n$ taken from a population with distribution $N(\mu,\sigma^2)$, whose average $\mu$ is unknown; [through the central ...
2
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1answer
45 views

Prove that the product of 2 vectors Normally distributed converges for large dimensions to the full zero matrix

Let $\mathbf{x}, \mathbf{y}$ $\in C^{M \times 1}$ are two i.i.d. vectors with distribution $\mathcal{CN(0,1)}$. How we can prove by the strong law of large numbers that: $\lim_{M\rightarrow \infty} ...
0
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1answer
18 views

Expectation of normal and log normal distribution

Let $X \sim N(\mu_x, \sigma_x^2)$ and $Y\sim N(\mu_y, \sigma_y^2)$, with correlation $\rho$. How do I find $$E[Xe^Y]$$? I tried a bunch of things without result. I'm also interested in "general" ...
0
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0answers
11 views

pseudo-Wishart distribution with shifted rows

I have a problem and I don't know where to start finding a solution. The problem is that I have a vector of i.i.d normal random variables such that, ...
0
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1answer
21 views

Properties of a distribution function

I'm having trouble understanding the properties of a distribution function. My book only gives these short rules. http://www.pixhost.org/show/2720/28297379_2015-06-22-15-27-44.jpg My professor said ...
0
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1answer
39 views

Correlation coefficients of X and Y [closed]

I was wandering if anybody could help me with the following question. I am fairly new to correlation coefficients and was attempting to tackle this question but was unsure how to do so? Thanks.
1
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1answer
51 views

To find $\sigma$ of a normal distribution

Given $X \sim \mathcal{N}( n, \sigma^2)$. The question told me $\mathbb{P}(X\lt 3) = \mathbb{P}(X\gt 7)$ So I found $n$ which is $5$. I'm also given $2\mathbb{P}(X\lt 2) = \mathbb{P}(X\lt 8)$. ...
0
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2answers
41 views

Explaining the form of the Gaussian measure

The Gaussian density $\mu(dx)=e^{-x^2/2}\ dx$ is fundamental in probability theory. Does anyone have a (non-computational) heuristic why this function should be special? (By non-computational, I mean ...
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0answers
22 views

How do I add uncertainties?

I have a gas for which we are continuously measuring composition with an online instrument. This composition is then being used to calculate some properties (dewpoint), and I want to estimate the ...
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0answers
10 views

Dice probability normal distribution

You roll a dice 1000 times. Calculate the probability you roll a six between 150 and 200 times. I understand how you calculate this with the binomial distribution: $$ = Binomialcdf(1000, 1/6, 200) - ...
5
votes
1answer
109 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
0
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1answer
23 views

A convergent sequence of normal random variables

Say $\{X_n\}$ is a sequence of normal random variables with means $0$ and variances $\sigma_n^2$. Also suppose that $X_n\to X$ (everywhere) and $\sigma_n^2\to \sigma^2.$ Then, using characteristic ...
0
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0answers
29 views

Equally spaced numbers with interval

I've this very silly doubt in getting equally spaced numbers with fixed interval from a set of numbers. Let's say I have a set of points, if I want maximum N points from this set equally distanced, ...
0
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0answers
29 views

Probability involving Normal Distribution

Two instruments are used to measure the height, h, of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056h. The error made by the ...
2
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1answer
21 views

Expectation of cumulative distribution function of a standard normal distributed random variable

Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. The find the expectation of $\Phi(X)$. I ...
0
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1answer
37 views

How to calculate $E(\sin^2X)$

If $X \sim N(0,1)$ then calculate $E(\sin^2X)$ I understand that $0 < \sin^2x<1$. So the expectation exists. I proceed as $E(\sin^2X)= \int_{-\infty}^{\infty}\sin^2xf(x)\,dx=2 ...
0
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1answer
31 views

How to calculate expected value of normal distribution with the condition that value is higher than x

I have following problem. Let assume that lifespan in the population has normal distribution with certain mean, variance and skewness. When the baby is born, its average lifespan will be equal to ...
0
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1answer
32 views

Show that for smaller $n$ than $n = 125000$ holds: $\mathbb{P}(|Z_n - \frac{1}{2}| \geqslant 0,01) \leqslant 0,02$

I'm a first year math student and I am having trouble with this exercise: Let $S_n$ be the amount of times we get heads when throwing a coin $n$ times. Let $Z_n = \frac{S_n}{n}$. With the equality ...
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2answers
43 views

How should I interpret this notation?

I am reading some lecture notes and I'm not sure how to interpret this: $$ b_j(x)=p(x\mid s_j)=N(x;\mu,\sigma^2)$$ It is clear from the context that $N$ refers to normal distribution, but what exactly ...
1
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0answers
21 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...
0
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1answer
36 views

Normal distribution for bags of coal produced from a machine.

A machine is used to bag coal, the mass of coal delivered per bag being normally distributed with mean 55 Kg and standard deviation 1.25 Kg. Given two filled bags chosen at random calculate the ...
1
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2answers
29 views

Teasing apart an explanation of the Central Limit Theorem

I'm looking at the central limit theorem, and cannot see in the explanation given to me how the average of identical distributions results in the normal distribution. I am told to consider a sequence ...
2
votes
1answer
38 views

What is the best way to interpolate over the 25th and 75th percentile of SAT scores?

The problem: I know the 25th and 75th percentiles of SAT scores for students admitted to a given university, and I want to interpolate over those two points in order to estimate all the percentiles ...
1
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2answers
53 views

PDF of $Z=\frac{X^2+Y^2}{2}$ where $X\sim N(0,1)$ and $Y\sim N(0,1)$

Say $X \sim N(0,1)$ and $Y\sim N(0,1)$ are independent random variables. So: $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}x^2}$ and $f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}y^2}$. Now I am ...
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0answers
92 views

Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
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0answers
25 views

Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
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0answers
16 views

Sum of Two Normal Distributions - Weighted?

The problem is given as such: John owns a portfolio with two stocks, ABC and XYZ. He has invested \$400 in ABC and \$600 in XYZ. The quarterly return on ABC is normally distributed with a mean of 7% ...
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0answers
26 views

Obtaining the standard deviation from a pair of (truncated) normal distributions

The expectation value of one side truncated (upper tail) normal distribution is defined as follows: $$ \operatorname{E}(X \mid X>a) = \mu +\sigma\lambda(\alpha) \!$$ where $$ ...
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1answer
23 views

central moment of normal distribution

In the normal distribution mean=2 &variance=4 then, 4th central moment is How to find out 4th central moment I solve through normal variate
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1answer
55 views

Is this PMF or PDF?

I am reading a technical report on expectation-maximization (EM) algorithm (http://melodi.ee.washington.edu/people/bilmes/mypapers/em.pdf) and I am confused about something. For HMMs, it defines ...
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1answer
24 views

Finding the mean of normal distribution through integration over $[a, b]$

I know the formula to finding this mean is to integrate $x\frac{1}{b}$ from $a$ to $b$. Can someone explain why this is so? I've been trying to compute the mean with the standard formula ($\int_a^b ...
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1answer
36 views

For $A,B,C$ independent and normal, what is $I(A+B;\ A+C)$?

Say $A,B,C$ are mutually independent and normally distributed with zero mean but possibly different variances $\sigma_1,\sigma_2,\sigma_3$. What is the mutual information between $A+B$ and $A+C$? All ...
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2answers
30 views

Central Limit Theorem and Normal Distribution problem.

Suppose I have a sample of people of size $n$ in which the probability that one smokes is p. I am asked what n should be so that the proportion of smokers in the samples is, in approximation of 0.01, ...
1
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1answer
34 views

Conditional density, bivariate normal

Let $Z=X+Y$ where $X \sim N(\mu,\sigma^2)$ and $Y \sim N(0,1)$ are independent. What is the conditional density of X given Z, $f_{X|Z}(x|z)$? I already found that ...