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

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2answers
63 views

Noise pdf Gaussian

Why the probability distribution function of the noise in a channel is Gaussian (normal distribution)? Intuitive discussion is appreciated.
1
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1answer
56 views

Finding confidence interval for a binomial process using the normal distribution?

See, when I was taught how to find confidence intervals, I always needed the sample variance to use a Student $t$ distribution to form the confidence interval. How does this work in the binomial case ...
1
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2answers
1k views

Mean and variance of the product of a normally distributed random variable

If a random variable X is normally distributed: $X \sim N(\mu,\sigma^2)$ what is the mean and variance of the random variable $Y = aX + bX^2$, where $a$ and $b$ are constant Given that ${\mathbb E} ...
2
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1answer
334 views

Going from the Poisson distribution to the Gaussian.

In this lecture, at about the $37$ minute mark, the professor explains how the binomial distribution, under certain circumstances, transforms into the Poisson distribution, then how as the mean value ...
1
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0answers
49 views

Calculating Expectation

I want to verify the following equation: $$E[(xe^{aY-\frac{1}{2}a^2}-b)^+]=x\Phi(l_1)-b\Phi(l_2)$$ where $Y\sim \mathcal{N}(0,1)$, $\Phi$ the distribution function of a standard normal distribution, ...
0
votes
1answer
400 views

Why do we use a $z$-test rather than a $t$-test when estimating an appropriate sample size?

I'm kinda puzzled on one point. In our stat class, we are taught to use the Student $t$ distribution to find confidence intervals for normally distributed data, as blindly using the normal ...
1
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1answer
241 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
3
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1answer
629 views

Convergence of binomial to normal

Problem: Let $X_n \sim \operatorname{Bin}(n,p_n) $ where $p_n \xrightarrow{} 0$ and $np_n \xrightarrow{} \infty$. What I need to show is that $$\frac{X_n - np_n}{\sqrt{np_n}} \xrightarrow{d} N(0,1) ...
1
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0answers
101 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
2
votes
1answer
255 views

3-D generalization of the Gaussian point spread function

I would like to extend to 3-D the formulation of the 2-D Gaussian PSF, given by: $$k_{\sigma}(x,y)=\frac{1}{\sqrt{(2\pi)^2}\sigma^2}\exp\left[-\frac{x^2+y^2}{2\sigma^2}\right]$$ Is the following 3-D ...
1
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2answers
112 views

Lower bound on the probability that the maximum of a sequence of $n$ i.i.d. standard normal r.v.'s exceeds $x$

Let $X_{\max}=\max(X_1,X_2,\ldots,X_n)$ where $n$ is large and each $X_i$ is i.i.d. standard normal random variable, i.e. $X_i\sim\mathcal{N}(0,1)$. Is there a lower bound on the probability ...
3
votes
2answers
53 views

Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
1
vote
3answers
421 views

Why is normal distribution more accurate than binomial distribution?

I'm having a tough time understanding this. This is what I am told about comparing the two: The probability that Saredo is late for school is 0.6. What is the probability that in one month she is ...
1
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1answer
42 views

Normal Distribution and a Discrete Amount

From what I understand about normal distribution is that you make a discrete number continuous by adding .5 which every way the question asks for. What if you were to have a discrete number with a ...
3
votes
1answer
216 views

Convergence in distribution and standard normal distribution

Let $X_1,X_2,\ldots$ be independent random variables with $X_k$ distributed as $\mathcal{N}(0,1)$ and $S_n=X_1 X_2+X_2 X_3+\ldots+X_n X_{n+1}.$ Show that $\frac{S_n}{\sqrt{n}}$ converges in ...
0
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0answers
79 views

Proof of theorem, multivariate normal distribution

The following theorem was presented in my textbook without proof and I would be thankful if someone could refer me to a proof of it: Suppose that $ \boldsymbol{X} \sim ...
1
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1answer
413 views

What is the difference between distribution and dispersion?

I need to explain the difference between a distribution (Normal, Chi-square, Poisson, etc.) and Dispersion (as measured by variance, standard deviation) to some students. What is the simplest ...
1
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1answer
74 views

How to normalize histogram well?

UPDATE 2 The question may be formulated as follows: Is there any common probability distribution, like normal distribution, but which has sharp (or just sharper) edges? If yes, then I could ...
0
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1answer
109 views

Normal random Vector

Question: Prove that linear functions of the form $\bar{y}=\bar{b}+\mathrm{B}\bar{x}$ are normal random vectors provided that $\bar{x}$ is a normal random vector. Find $E(\bar{y})$ and $V(\bar{y})$. ...
0
votes
1answer
152 views

Find the standard deviation of $ \frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$

Given $\frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$ as a normal distribution PDF with mean $\mu$, I'd like to solve for the std deviation in terms ...
1
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1answer
167 views

Normal Distribution Probability

At a large publishing company, the mean age of proofreaders is 36.2 years, and the standard deviation is 3.7 years. Assume the variables are normally distributed. a. If a proofreader in the company ...
1
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1answer
182 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
8
votes
2answers
227 views

Computing the Gaussian integral with step functions

Say, we are interested in deriving $$\int_{-\infty}^{\infty}e^{-x^2}=\sqrt{\pi}\tag{1}$$ There are many well known ways to do it, for example: by polar coordinates via the gamma function, etc. ...
1
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1answer
96 views

To what extent the statement “Data is normally distributed when mode, mean and median scores are all equal” is correct?

I read that normally distributed data have equal mode, mean and median. However in the following data set, Median and Mean are equal but there is no Mode and the data is "Normally Distributed": $ 1, ...
5
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3answers
275 views

Central Limit Theorem Definition

My friend and I have a bet going about the definition of the Central Limit Theorem. If we define an example as a number drawn at random from some probability density function where the function has a ...
5
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1answer
213 views

A case of the central limit theorem

I want to show that $$\frac{\sum_{k=1}^N X_k}{\sqrt{\sum_{k=1}^N X_k^2}} \overset{N\to\infty}{\to} \mathcal{N}(0,1)\text{ in distribution,}$$ where $X_1,X_2,\ldots$ is a sequence of iid random ...
4
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0answers
128 views

Characterization of the law of a stochastic process by its finite dimensional distributions

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space. Let $(X_t)_{t \in [0,T]}$, $(Y_t)_{t \in [0,T]}$ (real-valued) centered Gaussian processes such that the finite dimensional distributions ...
3
votes
2answers
182 views

$X,Y$ i.i.d., $X$ and $(X+Y)/\sqrt{2}$ have the same dist., then show that $X$ has a normal distribution

I am trying to show that $X$ is a standard normal (in distribution) by applying the Lindberg's version of the central limit theorem to a sequence always equal to $X$. In order to do that, I need to ...
1
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1answer
238 views

Projections of multivariate normal distribution

Given a random vector X with the multivariate normal distribution F(X), we know that, for two vectors a and b, the projections $A=\sum_j a_j X_j $ and $B=\sum_i b_i X_i $ are univariate normal. I'm ...
0
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1answer
73 views

Regression vs. Normal Distribution

I have to estimate something using historical data. Should I find the equation of the curve of best fit to estimate? Or use a confidence interval, standard deviation, and a z-score to calculate it? ...
2
votes
2answers
151 views

normal distribution, two independent random variable

if $X$ and $Y$ are independent normal distribuited random variables and $T=2X-Y-1$ and $E[X]=E[Y]=1$ and $Var(X)=Var(Y)=4$, what is $Var(T)$? I get $E[T]=E[2X-Y-1]=2-1-1=0$, but i don't know how to ...
2
votes
0answers
32 views

The distribution of the result of Monte-Carlo method

For example, if I want to determine the probability of getting tails when tossing a coin. By Monte-Carlo method, I toss the coin 1000 times and got 600 tails. As I know the distribution of the result ...
1
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0answers
73 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
1
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1answer
462 views

Relation chi-square and student-t distribution

First I want to prove that the sum $Y_1+...+Y_n$ where $Y_i=X_i^2$ and $X$ is standard normally distributed has density $f_n(x)=c_n x^{n/2-1}e^{-x/2}1_{x>0}$ I do not want to derive it, I would ...
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1answer
48 views

distribution of Brownian Motion involving integral

What is the distribution of $\int_{t}^{T} W(s)ds$? Given that W(t) is brownian motion. So far, I have the following, $\int_{t}^{T} W(s)ds$ = $(T-t)W(t) + \int_{t}^{T} (T-s)dW(s)$ Also, ...
1
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1answer
48 views

Normal distribution Q

Human heights are one of the many biological random variables that can be modelled by the normal distribution. The average height of Canadian women aged 18 and older is 163cm, while the average height ...
1
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1answer
28 views

Changing bounds of integrals

If I have: $$\int_{L}^{\infty }e^{\dfrac{-(x-\sigma \sqrt{T})^2}{2}}\,dx$$ let $y = x - \sigma \sqrt{T}$ $$\int_{L - \sigma \sqrt{T}}^{\infty }e^{\dfrac{-y^2}{2}} \, dy$$ Why does the lower bound ...
4
votes
1answer
84 views

Is there a real number, that is proven to be normal in every base, whose digits can be enumerated by an algorithm?

To clarify, I mean every natural number base $b$ where $b \geq 2$. If so, what is the algorithm to generate the number (and what is the number, if it has a name)?
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1answer
580 views

probability of sample variance lying between given values

Let $X_1,\ldots,X_n$ be a random sample of size $n = 10$ from a population which is Normally distributed with mean $= 48$ and variance $= 36$. What is the probability that the sample variance of such ...
2
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1answer
87 views

Square of a normal distribution

Let $Z$ have a normal distribution with mean $\mu$ and variance $1$. Show that $Z^2$ is a continuous random variable and find its p.d.f. I really don't know what to do with this... I tried working ...
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1answer
64 views

Importance of estimating $\sigma^2$ in linear Statistical model

Statistical model for Complete Randomized design $y_{ij} = \mu + \tau_i + \epsilon_{ij}$ where, $i$ denotes treatment and $j$ denotes observation. $i=1,2,...,k\quad and \quad j=1,2,..., n_i$ ...
0
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1answer
125 views

Conditional probability of the sum of r.v.

I have $n$ independent random variables $X_i$ with known PDF and CDF (say, Normal, but not necessarily with the same parameters). Given $U_1, U_2 \subseteq \{1,...,n\} $ such that $U_1 \cup U_2 = ...
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2answers
94 views

How do you get hyperbola by squaring a bell?

This is a Gaussian bell (aka normal distribution). Its square, I belive looks the same. Yet, I see that chi-square distribution, which is a sum of k such bell squares, looks like Take a look at ...
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1answer
300 views

Computing the expected value of a matrix?

This question is about finding a covariance matrix and I wasn't sure about the final step. Given a standard $d$-dimensional normal RVec $X=(X_1,\ldots,X_d)$ has i.i.d components $X_j\sim N(0,1)$. ...
1
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1answer
107 views

Integral involving normal densities

I am trying to solve the integral $$I(y)=\int_{\mathbb R}f(x,y)g(x)dx,$$ where $f(x,y)$ is the bivariate normal density with known mean $(\mu_1,\mu_2)$ and covariance matrix $\Sigma$ , and $g(x)$ is ...
0
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0answers
82 views

Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin

I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
0
votes
1answer
28 views

Splitting multivariate normal into individual (correlated) components

I have a multivariate normal variable $X$ with mean zero and variance $\Sigma$. I would like to write every component $X_i$ of $X$ as: $$ X_i = \phi_i Z_i $$ where $\phi_i$ is a scalar and $Z_i$ is a ...
0
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1answer
274 views

How do I prove Poisson appraches Normal distribution

I want to prove why the mean and variance of a $\operatorname{Poisson}(\lambda)$, is different when the time index approaches infinite (it's approximated by the mean and variance of a Normal). For ...
0
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0answers
164 views

Central Limit Theorem for Dependent Non-Identical Random Variables.

If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$. How do we apply ...
2
votes
2answers
317 views

Distributing $m$ balls into $n$ urns with no urn left empty. [duplicate]

If $m \geq n$, how many different ways are there of distributing $m$ indistinguishable balls into $n$ distinguishable urns with no urn left empty? I have no idea how to even start with this so any ...