Look at the time series plot, how to tell if the data is “random” and nonnormal or not?

I tried to solve this exercise in the book of “Time Series Analysis with Application in R”: Simulate a completely random process of length 48 with independent, chi-square distributed values, each with 2 degrees of freedom. Display the time series plot. Does it look “random” and nonnormal? Repeat this exercise several times with a new … Read more

Team Expected Points Given Very Limited Information

Suppose we were interested in projecting Team A’s expected points in a given game. Lets say, on average over a very large sample size, they score 3 points a game against an average team. Then lets say that Team B allows 2 points a game against an average team. Finally, we will say that the … Read more

Random Variables Study Example

I was studying random processes and couldn’t solve this? Could you help? “Let x and y be zero-mean, jointly Gaussian random variables. Assuming that Var(x)= σ2xand Var(y)= σ2y, find a scalar a in terms of variance of x and and rxy such that x-ay and y are independent random variables?” I don’t understand this: if … Read more

min/max and probability distributions

We have two identically distributed, independent, uniform variables on interval [0,1] : X1, X2. And Y1=max, Y_{2}=\min(X_{1},X_{2}). I want to find distribution f(y_{1}|y_{2}) and f(y_{2}|y_{1}). So I tried to approach it like this: firstly try to calculate distribution of f(y_{1},y_{2}=w) as a derivative of: F_{Y_{1},Y_{2}}(y_{1},y_{2}=w)=\mathbb{P}([{X_{1}}\leq y_{1} \wedge X_{2} \leq y_{1}]\wedge [X_{1} \leq w\ \vee X_{2}\leq … Read more

KL divergence between function of two distributions

Suppose we have two random variables: $X \sim f(x)$ and $X’ \sim f'(x)$ and we know how to compute $KL(f||f’)$. We apply a one to one function $g$ on both random variables which gives us two new variables: $Y \sim h(y)$ and $Y’\sim h'(y)$. Is there any closed from relation for $KL(h||h’)$ based on $KL(f||f’)$? … Read more

What is meant by “contiguous alternative hypothesis”?

I am reading a paper on Chimeric Alternatives which introduces what it calls contiguous alternatives. Namely, we have a sequence of random variables $[X_i]_1^n$ which follow some distribution $f$ under the null hypothesis, and we wish to test this against a “class of contiguous alternatives” where the distribution of each $X_i$ is some $\tilde{f}=\tilde{f}_n$ which … Read more

Algebra on random variables

I have the feeling this should be doable, or at least have an approximation, but I’m failing to find one. Let’s consider a random variable C, that belongs to a Truncated Exponential distribution between 0 and 1. If we observe n i.i.d. variables C, what is the distribution of the harmonic mean of this set? … Read more

Conditional cdf

I want to know that how conditioning will affect the CDF of dependent random variables. More specifically, let’s suppose, $\Gamma_R={g\over A}$ and $\Gamma_D={g\cdot h\over B}$, where $g$ and $h$ are exponential random variables and $A$ and $B$ are some constant values. I want to find $$\Pr \{min(\Gamma_R,\Gamma_D)<\gamma\}\quad\quad (1)$$ Since both $\Gamma_R$ and $\Gamma_D$ are statistically … Read more

MLE of the difference of multivariate normal populations?

I have two multivariate normal populations: $\boldsymbol{A_1}$~$N_r(\boldsymbol{\mu_1},\Sigma)$ with $n_1$ observations, and $\boldsymbol{A_2}$~$N_r(\boldsymbol{\mu_2},\Sigma)$ with $n_2$ observations. I am wondering if there is some way to express the likelihood of $\boldsymbol{A_1}-\boldsymbol{A_2}$. Separately, I know the likelihoods of $\boldsymbol{A_1}$ and $\boldsymbol{A_2}$ are: $L(\boldsymbol{A_1})=L(\boldsymbol{\mu_1},\Sigma)=(2\pi)^{-\frac{1}{2}pn_1}|\Sigma|^{-\frac{1}{2}n_1}exp{[-\frac{1}{2}\sum_{\alpha=1}^{n_1}(\boldsymbol{A_{1,\alpha}-\mu_1})^T\Sigma^{-1}(\boldsymbol{A_{1,\alpha}-\mu_1})]}$ and $L(\boldsymbol{A_2})=L(\boldsymbol{\mu_2},\Sigma)=(2\pi)^{-\frac{1}{2}pn_2}|\Sigma|^{-\frac{1}{2}n_2}exp{[-\frac{1}{2}\sum_{\alpha=1}^{n_2}(\boldsymbol{A_{2,\alpha}-\mu_2})^T\Sigma^{-1}(\boldsymbol{A_{2,\alpha}-\mu_2})]}$, respectively. The end goal is to find the MLEs of $\boldsymbol{A_1}-\boldsymbol{A_2}$ (both the … Read more

Polar Coordinate Variable Transformations

I have seen this question be touched on tangentially in many posts, but there does not seem to be one distinct place that asks about it directly. The following statement seems to be true, but I have not found a formal proof anywhere: X=Rcosθ, Y=Rsinθ are independent standard uniform random variables if and only if … Read more