## Prove that a symmetric distribution has zero skewness

Prove that a symmetric distribution has zero skewness. Okay so the question states : First prove that a distribution symmetric about a point a, has mean a. I found an answer on how to prove this here: Proof of E(X)=a when a is a point of symmetry Of course I used method 2 But now … Read more

## Probability of something happening exactly x times given y tries

I’m working on my stats homework, and I completely forgot how to do this. Also, we were only taught how to do it on a calculator (TI84) and I want to know how to do it without one. The problem is pretty much “what is the problem of it happening exactly twice given we try … Read more

## Link between exponential distribution and poisson probability mass function

Customers arrive randomly and independently at a service window, and the time between arrivals has an exponential distribution with a mean of 12 minutes. Let X equal the number of arrivals per hour. What is P[X = 10]? Now the solution to this problem uses this logic: If the time between arrivals is exponential with … Read more

## What’s the difference between MCMC and particle MCMC?

Markov chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a probability distribution based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. So how does the ‘particle’ bit augment the ‘MCMC’ bit? Answer Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which … Read more

## Proof Chebyshev’s Inequality

I want to proof Chebyshev’s Inequality using Markov’s inequality. Cheb.Ineq: P(|X−μ|≥a)≤Var(X)a So I’m starting with Markov’s Inequality: P(|X−μ|≥a)≤E(|X−μ|)a I replace |X−μ| by (X−μ), and square both sides, which leads to: P((X−μ)2≥a2)≤E((X−μ)2)a2 The numerator of the right hand side is the variance of X. So we get: P((X−μ)2≥a2)≤Var(X)a2 So far so good. The proof that I … Read more

## Interpreting OLS Regression Coefficients with High Multicolinearity

I am having trouble understanding the interpretation of OLS coefficients when predictors are highly correlated. My understanding of OLS coefficients is that they estimate a change in the expected outcome following a 1 unit increase in the predictor, holding all other predictors constant. However, I cannot understand why this doesn’t lead to underestimates following an … Read more

## How does one find the density of the kkth ordered statistic?

Let X1,…,Xn be n iid random variables. Suppose they are arranged in increasing order X(1)≤⋯≤X(n) The first ordered statistic is always the minimum of the sample Y1≡X(1)=min For a sample of size n, the nth order statistic is the maximum, that is, Y_n \equiv X_{(n)}=\max\{\,X_1,\ldots,X_n\,\} According to Wolfram Mathworld (here), if X has a probability … Read more