The moment-generating function of the gamma distribution is \( M(t) = (1-\theta t)^{-\alpha}\). Such a friendly little guy. Not what… Read more
Month: August 2011
Deriving the gamma distribution
Let’s take a look at the gamma distribution: $$ f(x) = \frac{1}{\Gamma(\alpha)\theta^{\alpha}}x^{\alpha-1}e^{-x/\theta}$$ I don’t know about you, but I think… Read more
The forgetful exponential distribution
The exponential distribution has the quirky property of having no memory. Before we wade into the math and see why,… Read more
Deriving the exponential distribution
The exponential distribution looks harmless enough: $$ f(x) = \frac{1}{\theta}e^{-x/\theta}$$ It looks like someone just took the exponential function and… Read more
Pointwise confidence interval estimates of survivor functions
The most commonly used descriptive method for survival data is the Kaplan-Meier estimator (also known as the product-limit estimator). It… Read more
Stabilization simulation of relative frequency
One of my favorite simulations is seeing how the relative frequency of an event stabilizes around a theoretical probability as… Read more
Simulating outcomes of a simple experiment
One of the first things you learn in an introductory mathematical statistics class is how to assign a probability model… Read more
Investigating the chi-square test of independence
Survey a random sample of 500 men and ask if they have seen Star Wars more than twice. Survey 500… Read more
Explaining and simulating a chi-square distribution
Randomly select several values, say 10, from a standard Normal distribution. That’s a Normal distribution with a mean of 0… Read more
Why the moment-generating function works
In a previous post, I described how a moment-generating function does what its name says it does. Still, though, it’s… Read more