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

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2
votes
0answers
118 views

Getting a Hermite polynomial expansion of Gaussian with given variance.

I am trying to find an expansion of centered Gaussian - $\frac{1}{\sqrt{2\pi}\sigma}\exp({-\frac{x^2}{2\sigma^2})}$ in terms of Hermite polynomials. Namely to calculate ...
1
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3answers
3k views

Normalizing a Gaussian Distribution

Assuming a Gaussian distribution with mean of zero and standard deviation of one, I would like to normalize this for an arbitrary mean and standard deviation. I know you're supposed to add the mean ...
1
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1answer
381 views

Finding distribution of $X^2+Y^2$ where $X,Y\sim N(0,1)$

Assume I have two random independent standard normal variables $X,Y\sim N(0,1)$, How can I find the distribution of $Z=X^2+Y^2$? I thought integrating the convolution, i.e ...
0
votes
0answers
332 views

Question about variance and expected value of $X\sim N(\mu,\sigma^2)$

I have a random variable $X\sim N(\mu,\sigma^2)$. Assume I define $Y=|X|$ , How can I find Y's variance and expected value? First I calculated its density directly and if I'm not mistaken it's ...
5
votes
3answers
157 views

How to Make a PDF 'Look' Uniform?

Let $X$ be a normally-distributed random variable with mean zero and variance $\sigma^2$: $X \sim N(0,\sigma^2)$. Let $Y$ be a mapping from $X$ onto the interval $(0,1)$ using the sigmoid function: ...
1
vote
1answer
82 views

Finding 'symmetrical range' from mean.

A machine used to make butter where its masses are normally distributed with mean m and standard deviation s.It is found that 5% from these butters are having mass more than 85g where else 10% are of ...
1
vote
1answer
79 views

Calculating Probability based on previous Probability values

So I am practising come exam questions on Normal distributions. I have this question which I cant quite get right. X,Y are normally distributed. Given X-> mean=100, variance=25. Given Y-> mean=120, ...
1
vote
1answer
2k views

Normal distribution with absolute value

I am new to the normal distribution topic. While I have understood and solved various different kind of questions, the normal distribution questions with absolute value, are the ones I have no idea ...
1
vote
2answers
126 views

Upper Limit in normal distribution?

(Iv already solved the a) part with the answer 0.2119, which is correct. The b) part asks for the upper limit, I dont know what an upper limit is in these type of questions. Can any one give me ...
2
votes
2answers
47 views

Mean and standard deviation with percentages

(I am very new to this topic and I have tried many questions successfully. Except story questions like these, which confuse me. Can I just get hints that help me out instead of the answer? or like ...
3
votes
1answer
68 views

Length of Gaussian distributed variables

Suppose I have a set of random variables $x_1,...,x_n$ s.t. $x_i\sim N(\bar{x}_i,\sigma_i^2)$. And I define a new variable $x=\sqrt{x_1^2+...+x_n^2}$, then will $x$ also be normally distributed? And ...
1
vote
1answer
110 views

Solve covariance matrix of multivariate gaussian [duplicate]

This is a practical, and basic question. I have a multivariate Gaussian in $M$ dimensions with center $\mu$ (known, lets assume $0$) and some points $p$ where I have the value of $$ \ln(L)= ...
2
votes
1answer
48 views

Maneuvering normal distribution inequality

I know $\Pr[N(\mu, \sigma^2) \geq \mu + k \sigma] = \Pr[N(0, 1) \geq k]$. Say I am given, $\Pr[N(\mu, \sigma^2) \geq \mu + q \sigma] = \Pr[N(0, 1) \leq k]$. How can I find a relation between $q$ and ...
1
vote
0answers
131 views

Berry-Esseen Theorem-like result with fourth central moment instead of third absolute moment

Let $X_i$, $i=1,\ldots,n$ be i.i.d. random variables with $E[X_i]=\mu$, $E[(X_i-\mu)^2]=\sigma^2$, and $E[(X_i-\mu)^4]=\kappa$. I am interested in approximating the distribution of ...
3
votes
2answers
466 views

Does finite fourth moment imply finite third absolute moment?

This may be a silly question, but, for a random variable $X$ defined on reals, I am wondering if the existence of the finite fourth central moment $E[(X-E[X])^4]$ implies the existence of the finite ...
1
vote
0answers
26 views

Confusion related to predictive distribution of gaussian processes

I have this confusion related to the predictive distribution of gaussian process I didn't get how the integration gave that result. What is P(u*|x*,u). Also how come the covariance of the posterior ...
2
votes
1answer
36 views

Valid distribution

If $N(0,\sigma^2)$ is the Gaussian distribution with mean $0$ and variance $\sigma^2$, is $pN(0,\sigma^2)$ a valid distribution ? $p$ is a constant and $0\le p\le 1$.
2
votes
0answers
299 views

Distribution of the $l_2$-norm of gaussian vector

Let $Y_k \sim N(\mu_k, \sigma_k^2)$. For $\sigma_k = \sigma$ the squared norm of $Y = (Y_1, \ldots, Y_n)$ follows the noncentral chi square distribution. What is the distribution in the general case? ...
1
vote
1answer
123 views

Creating an offset bell curve

This is half programming and half math, but I need the math portion answered as I'm no good at it. I have a list of 10 objects and am randomly selecting and object from that list. I need the ...
0
votes
0answers
50 views

How $\mathbb E[\bar\epsilon_{i.}-\bar\epsilon_{..}]=0$ ? $\mathbb E$ denotes expectation.

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

How come $P(Z< -1.5)$ is equal to $P(Z > 1.5)$ which are both equal to $1-P(Z < 1.5)$?

I can't wrap my head around the idea they are both equal. I mean shouldn't we have $P(-Z > 1.5)$ which is not equal to $P(Z < 1.5)$?
3
votes
1answer
188 views

Help with gaussian integral

I need to solve this gaussian integral: $$\int_\mathbb{R} (2\pi)^{-n/2}\mid \Sigma\mid ^{-\frac{1}{2}}e^{-\frac{1}{2}(u-Kx)^T\Sigma ^{-1}(u-Kx)} u^TRu \,\mathrm du$$ It is the integral of a ...
1
vote
0answers
729 views

find probability in normal distribution

i would like to check myself if following my answer is correct: let us consider following problem: Suppose the heights of a population of $3,000$ adult penguins are approximately normally distributed ...
2
votes
1answer
31 views

Confusion related to matrix multiplication

I am having this simple confusion.Lets consider a multivariate gaussian distribution with mean $\mu$ and precision matrix $K$. Then the exponential term is $$(x-\mu)' K (x-\mu)$$ If I open the above ...
0
votes
2answers
203 views

Shift interval of log-normally distributed latin hypercube samples

first of all I'm not sure if this part of StackExchange is the right one because my question is mainly on a way to implement something in MATLAB. Ok, now let me try to pack my whole question in one ...
0
votes
1answer
616 views

How to merge two Gaussians

I have two multivariate Gaussians each defined by mean vectors and Covariance matrices (diagonal matrices). I want to merge them to have a single Gaussian i.e. I assume there is only one Gaussian but ...
0
votes
2answers
221 views

Probability Distribution. Case study with a bacterial population

Let's imagine, we start with one single bacterium. At each time step (generation), each bacterium has $x$ offspring and it dies (semelparous species). $x$ is a value drawn from a normal distribution ...
0
votes
1answer
114 views

Lower bound on the probability of maximum of $n$ i.i.d. chi-square random variables exceeding a value close to their number of degrees of freedom

I am wondering if there is a tight lower bound on the probability of a maximum of $n$ i.i.d. chi-square random variables, each with degree of freedom $d$ exceeding a value close to $d$. Formally, I ...
0
votes
1answer
28 views

Cumulative Normal Distribution.

Let $X_1,\ldots,X_n$ be a random sample from $f(X;\theta)=\phi_{\theta,25}$, that is, $X_1,\ldots,X_n$ be normally distributed with mean $\theta$ and variance $25$. I am not understanding how ...
2
votes
1answer
677 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
350 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
78 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
120 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
194 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
50 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
482 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
761 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
64 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
840 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
64 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
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
vote
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
votes
1answer
383 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
429 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 ...
2
votes
1answer
253 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
675 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
103 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
298 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 ...