Articles of normal distribution

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}’\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in N(\boldsymbol{\mu},\boldsymbol{\Lambda})$. We observe that $Z = \boldsymbol{t}’\boldsymbol{X}$ has a one dimensional normal distribution. the parameters are $m = E[Z] = \boldsymbol{t}’\boldsymbol{\mu}$ and $\sigma^2 = \mathrm{Var}[Z] = \boldsymbol{t}’\boldsymbol{\Lambda}\boldsymbol{t} $($\boldsymbol{\Lambda} $ is the covariance matrix). since $\textbf{(1)}\enspace \varphi_{\boldsymbol{X}}(\boldsymbol{t})=\varphi_{z}(1) = \exp\{im […]

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: >>> import numpy as np >>> from scipy import stats >>> a=np.random.random_integers(1,9,4) >>> a array([3, 7, 4, 3]) >>> b=np.random.random_integers(1,9,5) >>> […]

Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.

My question is: do you know any examples when $X$ and $Y$ are both normally distributed, but the two dimensional vector $(X,Y)$ is not? I found some example in book, but I don’t understand it. The example is: Let us assume that $f_1$ and $f_2$ are both the densities of $2D$ normal distribution with $0$ […]

Iteratively Updating a Normal Distribution

Is there a way to update a normal distribution when given new data points without knowing the original data points? What is the minimum information that would need to be known? For example, if I know the mean, standard deviation, and the number of original data points, but not the values of those points themselves, […]

If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random variables such that $X+Y$ is normal. Is it necessarily the case that $X$ and $Y$ are also normal? Thoughts: If the […]

Characteristic function of a product of random variables

I am facing the following problem. Let $X,Y$ be independent random variables with standard normal distribution. Find the characteristic function of a variable $ XY $. I have found some information, especially the fact that if $ \phi_X,\phi_Y $ denote characteristic functions, then $$ \phi_{XY}(t) = \mathbb{E}\phi_X(tY).$$ The only problem is that the proof required […]

$Y$ is normally distributed or constant if $X$ and $X+Y$ are normally distributed and $X$, $Y$ are independent

Let $X$, $Y$ be independent random variables with $X$ and $X+Y$ normally distributed. Prove that $Y$ is also normally distributed or is constant a.s.

Covariance between squared and exponential of Gaussian random variables

Assuming the random vector $[X \ \ Y]’$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]’$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X \ \rho \\ \sigma_Y \ \sigma_X \ \rho & \sigma_Y^2 \end{array} \right]$, I am looking for the expression of $\text{Cov}(X^2, \exp{Y})$. I know there […]

Expected value of $x^t\Sigma x$ for multivariate normal distribution $N(0,\Sigma)$

For standard normal distribution, the expected value of $x^2$ is $1$. A natural question is that in the multivariate case, what is the expected value of $x^t\Sigma x$ for multivariate normal distribution $x \sim N(0,\Sigma)$? I have difficulty to carry out the integral, but would guess the result is related to the norm of $\Sigma$.

Definite integral of Normal Distribution

Possible Duplicate: How to directly compute an integral which corresponds to the normal distribution Is there any approximate solution for the following definite integral of normal distribution? $$\int_{a}^{b} e^{-\frac{(x-\mu)^2}{2s^2}} \ dx$$