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Month: August 2011


The moment generating function of the gamma

Posted on August 30, 2011 Updated on August 18, 2023

The moment-generating function of the gamma distribution is \( M(t) = (1-\theta t)^{-\alpha}\). Such a friendly little guy. Not what… Read more

Posted in Continuous distributions, Expectation     Leave a comment     


Deriving the gamma distribution

Posted on August 29, 2011 Updated on July 7, 2024

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

Posted in Continuous distributions     8 Comments     


The forgetful exponential distribution

Posted on August 26, 2011 Updated on August 16, 2023

The exponential distribution has the quirky property of having no memory. Before we wade into the math and see why,… Read more

Posted in Continuous distributions, Probability     1 Comment     


Deriving the exponential distribution

Posted on August 25, 2011 Updated on August 16, 2023

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

Posted in Continuous distributions     8 Comments     


Pointwise confidence interval estimates of survivor functions

Posted on August 23, 2011 Updated on August 16, 2023

The most commonly used descriptive method for survival data is the Kaplan-Meier estimator (also known as the product-limit estimator). It… Read more

Posted in Survival analysis     Leave a comment     


Stabilization simulation of relative frequency

Posted on August 15, 2011 Updated on August 16, 2023

One of my favorite simulations is seeing how the relative frequency of an event stabilizes around a theoretical probability as… Read more

Posted in Simulation, Using R     1 Comment     


Simulating outcomes of a simple experiment

Posted on August 14, 2011 Updated on August 16, 2023

One of the first things you learn in an introductory mathematical statistics class is how to assign a probability model… Read more

Posted in Simulation, Using R     Leave a comment     


Investigating the chi-square test of independence

Posted on August 12, 2011 Updated on August 16, 2023

Survey a random sample of 500 men and ask if they have seen Star Wars more than twice. Survey 500… Read more

Posted in Continuous distributions, Probability     Leave a comment     


Explaining and simulating a chi-square distribution

Posted on August 7, 2011 Updated on August 16, 2023

Randomly select several values, say 10, from a standard Normal distribution. That’s a Normal distribution with a mean of 0… Read more

Posted in Continuous distributions, Simulation, Using R     Leave a comment     


Why the moment-generating function works

Posted on August 6, 2011 Updated on August 14, 2023

In a previous post, I described how a moment-generating function does what its name says it does. Still, though, it’s… Read more

Posted in Expectation     2 Comments     


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