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

learn more… | top users | synonyms

0
votes
1answer
56 views

Homework Help. Probability Density Functions.

$X$ is $N(10,1)$. Find $f(x|(x-10)^2 < 4)$ This is a homework question. I can only figure out that X is normally distributed with mean 10 and variance 1. Can you please explain what is meant to ...
2
votes
0answers
80 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
0
votes
1answer
63 views

Normal Distribution Calculating Probability

I am struggling with the following question: A company which produces $1L$ beverages adjusts their machines in a way that the filling quantity is normally distributed. The mean is ...
1
vote
2answers
119 views

Bound for erf function

For small $\epsilon \geq 0$ Is $erf(\epsilon) \leq \epsilon$ Can somebody give me the hint
0
votes
1answer
46 views

Bound for the integral

Is there any way to bound the following integral $$\int_{-(\epsilon+1)/\sigma}^{(\epsilon-1)/\sigma} \mathrm e^{-t^2/2}\,dt$$
1
vote
1answer
2k views

Derive the PDF of the log-normal distribution?

If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods: (i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
6
votes
0answers
277 views

What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
2
votes
1answer
187 views

Can one sample uniformly from the surface of an $n$-sphere of non-unit radius using normal r.v.'s?

One can sample coordinates of the surface of a unit radius $n$-dimensional sphere uniformly using the following method: independently generate a vector of $n$ standard normal random variables ...
0
votes
2answers
675 views

Indefinite integral of product of CDF and PDF of standard normal distribution

Is there a solution to: $\int ^\infty _x \Phi(z) \phi(z) dz$ where $z$ ~ $N(0,1)$ and $\Phi$ and $\phi$ refer to the CDF and PDF? Many thanks.
1
vote
1answer
88 views

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution

I am trying to derive Chi-square distribution. The random variale is $$ U^2=\sum_{i=1}^k X_i^2 $$ where $X$ is a random variable with normal standard distribution. What is the distribution of ...
2
votes
3answers
10k views

Standardizing A Random Variable That is Normally Distributed

To standardize a random variable that is normally distributed, it makes absolute sense to subtract the expected value $\mu$ , from each value that the random variable can assume--it shifts all of the ...
1
vote
2answers
468 views

Can Bhattacharyya distance be greater than one?

I have two vectors, say $P$ and $Q$. I want to find the statistical overlap between two given that $P$ is my reference which I have modeled after Normal distribution and I have parameters for it. $Q$ ...
-1
votes
1answer
22 views

Bound for normal distribution

suppose $X$ is a standard normal distribution then what is the bound for $Pr \{|X|\leq \epsilon \} $, where $\epsilon \geq 0.$
0
votes
1answer
153 views

special matrix in terms of its covariance matrix

How can we find a matrix $S\in \mathcal{M}_{n,n}$ and $Z\in \mathcal{M}_{n,m}$ whose $n$ entries of the $i^{th}$ column $Z_i$ are correlated $Z_i \sim \mathcal{N}(0,S)$ where $S \in \mathcal{M}_{n,n}$ ...
2
votes
1answer
1k views

Variance of $\exp(-x)$

Hi I have been struggling to find the variance of the $\exp(-x)$ in terms of $\exp$. For the function Y = exp (-x) where X is N (0,1) show that the variance of Y = $\exp(\exp-1)$ This is what I ...
3
votes
3answers
121 views

$E(1/(1+x^2)) $under a normal distribution

I want to know as mentioned in topic $E(1/(1+x^2))$ under a normal distribution $N(0,1)$. If it's not analytical, are there any bounds that are possible? So basically, ...
0
votes
1answer
102 views

Normal distribution bound

Let $X$ be a random variable which follows normal distribution. Is True that $Pr[|X|\leq \epsilon] \leq \epsilon$ for all $\epsilon \geq 0$.
0
votes
1answer
36 views

Normal distribution in equality

Let $p(x)=a_1x_1+a_2x_2+. . . .a_n x_n$ be a polynomial such that $\sum_ia_i^2=1,$ each $x_i \sim N(0,1)$. then we know that $p(x) \sim N(0,1)$. How can we bound $\Pr_{x\in ...
0
votes
2answers
74 views

probability normal distribution

A model for the movement of a stock supposes that if the present price of the stock is s, then after one time period it will be either (1.012)s with probability 0.52, or (0.99)s with probability ...
1
vote
1answer
63 views

Probability - normal distributions

The time it takes for a calculus student to answer all the questions on certain exam is an exponential random variable with mean 90 minutes. If all 100 students of a calculus class are taking that ...
3
votes
2answers
237 views

Covariance of a Normal with its Square

Assume there is a random variable distributed normal $X\sim N(\mu,\sigma^2)$. Is there an analytic expression for the covariance of $X$ with its square $X^2$? $$\operatorname{Cov}(X,X^2)$$ I have ...
0
votes
1answer
133 views

How to prove that $Y=\ln(X)$ approximately Normal when $X$ is a Normal random variable with $\mu\gg\sigma$

I wanted to prove that PDF of $Y=\ln(X)$ tends to a Normal distribution with $\mathcal{N}(\ln(\mu_{x}),\sigma^{2}_{y})$ when $X\sim\mathcal{N}(\mu_{x},\sigma^{2}_{x})$. It is also important to note ...
1
vote
0answers
130 views

Problem involving the bivariate normal distribution

If $X$ and $Y$ have a bivariate normal distribution with $\mu(x)=\mu(y)=0$, $\rho=0$, $\sigma(x)=\sigma(y)=10$. Find the following: A) The probability of getting a point $(x,y)$ inside the ...
1
vote
0answers
204 views

Hypothesis testing of normal distribution, known mean unknown variance

I've been working on review problems, and this one has me completely stumped. Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the ...
0
votes
0answers
63 views

Using an appropriate hypothesis to test whether two means are different

Manager examined potential differences between two models of bicycles. The mean life of the bicycles is of primary concern. The followings table provides the available date which measured in ...
8
votes
1answer
360 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
2
votes
1answer
90 views

what is the joint distribution of these two random variables?

what is the joint distribution of two random variables,$∑_iX_iY_i$ and $(∑_iX_i)(∑_jY_j)$? Note that since $n$ is a large number and all the random variables are $iid$, using central limit theorem, ...
-1
votes
3answers
339 views

Variance of transformed random variable

The relationship of two random variables is given by $$ X = \Phi(Y) \Leftrightarrow Y = \Phi^{-1}(X),$$ where $\Phi(\bullet)$ is the standard normal cdf and $\Phi^{-1}(\bullet)$ the inverse of the ...
1
vote
0answers
246 views

Bayesian posterior with integrals over normal densities

Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
1
vote
1answer
106 views

Does $0$ correlation imply independence for marginally normal distributions?

Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$. Can someone give a hint why this is true ?
1
vote
3answers
130 views

Let $ X_1,X_2,…,X_n$ be i.i.d. $N(\theta_1, \theta_2)$, please prove that $E[(X_1-\theta_1)^4] = 3\theta_2^2$

If $X_{1}$, $X_{2}$, ..., $X_{n}$ is sampled from $N(\theta_1, \theta_2)$, how can I prove that $E [(X_{1} - \theta_1)^{4}] = 3 \theta_2^{2}$? I started off this question finding the completely ...
-2
votes
2answers
285 views

Is first order moving average a Markov process?

Given first order moving average $$ x(n) = e(n) + ce(n-1) $$ where $e(n)$ is a sequence of Gaussian random variables with zero mean and unit variance which are independent of each other, and $c$ is ...
1
vote
1answer
43 views

Cumulative Distribution Function of $X = Y*I + Z*(1-I)$

I have a random variable $X = Y*I + Z*(1-I)$ where $Y~N(u, sigma^2)\,,\,\, Z~N(u, sigmac^2)$, and $I~B(1,1-p)$. What I can't seem to figure out is how to get a cumulative distribution function for ...
0
votes
1answer
45 views

Interval Estimation when $\overline{Y}$ and $S$ is unknown.

Question: A random sample of size $n=9$ is drawn from a normal distribution with $\mu=27.6$. Within what interval $(-a,+a)$ can we expect to find $\frac{\overline{Y}-27.6}{S/\sqrt{9}}$ $80$% of the ...
2
votes
1answer
208 views

Chance of two Gaussian distributions being observations of the same phenomenom?

For an algorithm used for generation of a road map based upon position-samples, I am looking for a method of determining the probability of a sample belonging to an already discovered element of the ...
1
vote
1answer
66 views

Independence of Combination of Normal Random Variables

$\newcommand{\Cov}{\operatorname{Cov}}$ I have a practice question I'm trying to answer in studying for an upcoming exam: $X\sim N(0,1)$ and $Y\sim N(0,1)$ and I have $\rho(X,Y)=0.4$. Define a ...
1
vote
0answers
39 views

algorithm to use to balance a set of IPs into a set of buckets

So we have a set of IP addresses (~3000) and want to balance them into 4 different buckets. What we are doing now is very simple by treating the last part of the IP as integer and mod it by 4. e.g. ...
0
votes
1answer
70 views

What does $E[{\bf{x}} {\bf{x}}^{T}]$ mean?

It's known that $E[{\bf{x}} {\bf{x}}^{T}]={\bf{\mu \mu}}^{T}+{\bf{\Sigma}}$ but I have seen a very similar identity using data points $\bf{x_{n}}$ and $\bf{x_{m}}$ sampled from a multivariate Gaussian ...
1
vote
1answer
262 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
2
votes
1answer
511 views

What is the analytic expression for PDF of joint distribution of two Gaussian random vectors?

I know that if $X$ and $Y$ are random variables with respective PDFs, $$ f_X(x) = \frac{1}{\sqrt{2\pi\sigma_x^2}}\exp\left\{-\frac{\left(x-\mu_x\right)^2}{2\sigma_x^2}\right\} \\ f_Y(y) = ...
2
votes
1answer
415 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
2
votes
1answer
2k views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
2
votes
1answer
109 views

Log-likelihood for multinominal normal distribution

Given $n$ jointly-normal random variables $X_1, X_2, \dots, X_n$, with $$\mu_i=\mu\forall i \in\mathbb{N}^+$$ $$\sigma_i=\sigma\forall i$$ $$\rho_{i,j}=\rho\forall i,j \mbox{ with } i\neq j$$ what is ...
0
votes
1answer
139 views

Histogram with Gaussian bell curve

How do I create/calculate the probability density curve in a histogram which is scaled to the frequency axis with ABSOLUTE values (example)? The curve should be based on the calculated average and the ...
0
votes
1answer
105 views

What is the distribution of an unconditioned random variable knowing the conditional distribution?

I have two random variables $X$ and $Y$. I know that $Y$ can be approximated by a $N(\mu_1,\sigma_1^2)$ distribution (in particular $Y$ is not negative) and I also know that $X|Y \sim N(a+bY,c+dY)$ ...
1
vote
2answers
596 views

The correlation between two normal distribution

Let $X$ have the $N(0,1)$ distribution and let $a>0$, show that the random variable $Y$ given by $$Y=\begin{cases} X & \text{if }|X|<a\\[5pt] -X &\text{if }|X|\geq a\; \end{cases}$$ has ...
2
votes
1answer
4k views

Prove Variance of a normal distribution is (sigma)^2 (using its moment generating function)

Prove that the Variance of a normal distribution is (sigma)^2 (using its moment generating function). What I did so far: Var(X) = E(X^2) - (E(X))^2 E(X^2) = Mx'(0) = r/(sqrt(2pi)*sigma) * ...
4
votes
1answer
194 views

Expectation value of $1/x$

Given a random variable $x$ which is assumed to follow a Gaussian distribution $x \sim N( \mu, \sigma^2 )$ and $x$ is further known to be positive, I am interested in the following expectation value: ...
1
vote
1answer
101 views

Mentally Estimating the Normal CDF

More than once I have seen this sort of frustrating question on a Mathematics GRE practice test: A fair die is tossed 360 times. The probability that a six comes up on 70 or more tosses is... a) ...