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

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2
votes
1answer
617 views

Expected Value of Normal CDF

I am trying to calculate the expected value of a Normal CDF, but I have gotten stuck. I want to find the expected value of $\Phi( \frac{a-bX}{c} )$ where $X$ is distributed as $\mathcal{N}(0,1)$ and ...
3
votes
0answers
88 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
4
votes
2answers
315 views

Distribution of $Y = \sin X$ when $X$ is normal?

Assume $X$ is Normally distributed : $X\sim N(\mu,\sigma)$ What is the distribution of $Y = \sin X$ ? I think we should start with $F_Y(y)=P(\sin X < y)$. But how to continue?
3
votes
0answers
76 views

Is there an algebraically normal function from $\mathbb{Z}^{2}$ to $\{ 0 , 1\}$?

Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve. Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r ...
2
votes
1answer
111 views

Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
2
votes
3answers
185 views

How to integrate the difference between the CDFs of two normal distributions

I have two normal distributions A and B. I am trying to write a program that will take mean(A), stddev(A), mean(B), stddev(B) and output the result of the following equation: $$ ...
1
vote
1answer
49 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
9
votes
3answers
452 views

Is the mean of the truncated normal distribution monotone in $\mu$?

I am wondering whether the mean of the truncated normal distribution is always increasing in $\mu$. The untruncated distribution of $x$ is $\mathcal{N}(\mu,\sigma^2)$. The mean of the truncated ...
3
votes
0answers
600 views

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
0
votes
0answers
63 views

iterative transform of standard normal random variable

Given a discrete series of random variable $n(i)$ that each element follows the standard normal distribution $N(0,1)$, another series is defined iteratively as: $$u(i+1)=au(i)+bn(i)$$ where ...
1
vote
0answers
740 views

calculate probability without table

my question is related to normal distribution,namely as i know in GRE quantity section,there could be question related to normal distribution,but of course we will not have table,o how can we ...
0
votes
2answers
62 views

Noise pdf Gaussian

Why the probability distribution function of the noise in a channel is Gaussian (normal distribution)? Intuitive discussion is appreciated.
1
vote
1answer
55 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
vote
2answers
976 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
votes
1answer
282 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
vote
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
371 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
vote
0answers
232 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
votes
1answer
600 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
vote
0answers
97 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
214 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
vote
2answers
109 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
404 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
vote
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
215 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
votes
0answers
74 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
vote
1answer
375 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
vote
1answer
71 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
votes
1answer
108 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
147 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
vote
1answer
156 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
vote
1answer
167 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
220 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
vote
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
votes
3answers
269 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
votes
1answer
211 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
votes
0answers
124 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
177 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
vote
1answer
224 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
votes
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
147 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
31 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
vote
0answers
72 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
vote
1answer
455 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 ...
1
vote
1answer
47 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
vote
1answer
47 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
vote
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)?
1
vote
1answer
530 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 ...