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 […]

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) >>> […]

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 $0$ […]

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, […]

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 […]

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 […]

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.

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 […]

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 $\Sigma$.

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? $$\int_{a}^{b} e^{-\frac{(x-\mu)^2}{2s^2}} \ dx$$

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