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

What is the probability that for exactly three calls the lines are occupied?

I am unable to understand the following statement. The phone lines to an airline reservation system are occupied 40% of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that 10 calls are placed to the airline. $\textbf{What is the probability that for exactly three calls the … Read more

Why the standard deviation of the sample mean is calculated as σ√n\frac{\sigma}{\sqrt{n}}?

According to Wikipedia, the standard deviation of a sample mean is calculated as follows σ√n Why is that? Why do we need to divide the standard deviation of the population by the square root of n (which should I think be the size of the sample)? Why should that make sense? Answer The sample mean … Read more

Example of using Delta Method

Let ˆp be the proportion of successes in n independent Bernoulli trials each having probability p of success. (a) Compute the expectation of ˆp(1−ˆp). (b) Compute the approximate mean and variance of ˆp(1−ˆp) using the Delta Method. For part (a), I can calculate the expectation of ˆp but got stuck on the expectation of ˆp2, … Read more

If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator?

If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator? Here’s what I tried, is this right? $$ \begin{align} \mathbb{E}[\hat{\theta}_1] &= \mathbb{E}[e^{-X}] \\ &= e^{\mathbb{E}[-X]}\\ &= e^{-u} \\ &= \theta \end{align} $$ Show that $\hat{\theta}_2 = w(X)$ is an unbiased estimator of $\theta$, where $w(0)=1$ and $w(x)=0$ if $x> 0$. … Read more