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As always, you can download the code for these applications here.
In fact, you can think about this section as kind of another story for the Beta: The Beta wears many hats, and one is that it is a conjugate prior for the Binomial distribution. Say that you were interested in polling people about whether or not they liked some political candidate.
This is where the Beta comes in. We gave the Normal distribution as an example for the prior simply because we are very familiar with it; however, it is probably clear why the Normal distribution is not the best choice for a prior here. The support of a Normal distribution covers all real numbers, and no matter how small you make the variance, there will always be a chance that the random variable takes on a value less than 0 or greater than 1.
Do any distributions come to mind?
Well, what about the Beta? Concentrate the Beta around there. Flatten the Beta out, all by simply adjusting the parameters. This gives the Beta an advantage over the other bounded continuous distribution that we know, the Uniform: You get the idea. Before we calculate this, there is something we have to keep in mind: This sounds strange, but bear with it for now.
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Reference this tutorial video for more; there is a lot of opportunity to build intuition based on how the posterior distribution behaves. This is the definition of a conjugate prior: The Beta allowed us to explore prior and posterior distributions, and now we will discuss a new topic within the field: For concreteness, you might assume that they are all i.
It helps to step back and think about this at a higher level. Recall that a function of random variables is still a random variable i. You could also think about the official definition of statistic: There are a couple of reasons for this simplification. Second, discrete random variables have probabilities of ties when we try to order them.
Recall that continuous random variables have probability 0 of taking on any one specific value, so the probability of a tie i. Think about what this probability is: This might look like sloppy notation, but recall that the probability that a continuous random variable takes on any specific value is 0, so we can ignore the edge case of equality.
This is the story of a Binomial! Using the CDF of a Binomial, we write:. Consider this in extreme cases. Does this make sense? We can confirm our CDF result with a simulation in R. The reason is that there is a very interesting result regarding the Beta and the order statistics of Standard Uniform random variables.
This is an intuitive result because we know that the Uniform random variables marginally all have support from 0 to 1, so any order statistic of these random variables should also have support 0 to 1; of course, the Beta has support 0 to 1, so it satisfies this property. Many statisticians consider this to be the story of a Beta distribution: This result also helps to justify why we called the Beta a generalization of the Uniform earlier in this chapter!
We can check this interesting result with a simulation in R. We generate order statistics for a Uniform and check if the resulting distribution is Beta. Fortunately, unlike the Beta distribution, there is a specific story that allows us to sort of wrap our heads around what is going on with this distribution.
The Gamma has two parameters: Recall the Exponential distribution: Also recall that the Exponential distribution is memoryless, so that it does not matter how long you have been waiting for the bus: The Gamma distribution can be thought of as a sum of i. Exponential distributions this is something we will prove later in this chapter.
What does this remind us of? Exponential random variables; specifically, we know of two at the moment. In this case, the latter is easier we will explore this later in the chapter, after we learn a bit more about the Gamma. And, thus, this is the mean and variance for a Gamma. The key is not to be scared by this crane-shaped notation that shows up so often when we work with the Beta and Gamma distributions. This function is really just a different way to work with factorials.
How did we get to this result?
It seems kind of crude, but this is the idea behind forging the PDF of a Gamma distribution and the reason why it is called the Gamma! We know that, by the transformation theorem that we learned back in Chapter We can actually apply this here: Why can we so quickly say that this is true? Well, the Gamma distribution is just the sum of i.
So, our sanity check works out. You can further familiarize yourself with the Gamma distribution with this Shiny App; reference this tutorial video for more. Hopefully these distributions did not provide too steep a learning curve; understandably, they can seem pretty complicated, at least because they seem so much more vague than the distributions we have looked at thus far especially the Beta and their PDFs involve the Gamma function and complicated, un-intuitive constants.
We will now discuss their relationship to each other, which is also very important recall that connections between random variables are often more interesting than the random variables marginally. The set-up is as follows: Both places are notorious for having lines that you have to wait in before you actually reach the counter.
These wait times are independent. This is an excellent problem to help us practice our work with transformations, as well as the Beta and Gamma distributions. Recall the formula for a 2-D transformation:. We can start to plug in:. Completing these derivatives yields:. We are left with:.
How many of these random variables do we have? We actually almost have this; if we group terms, we see:. Have you been paying attention to the term on the right? At this point, we have actually reached a couple of really interesting results.
We can confirm our results in R by simulating from the relevant distributions. Again, the most important thing to take away from Bank-Post Office is strictly the result, which is listed out above.
Hayden Grant is starting her sophomore year and has one major goal in mind - to become a member of Beta Gamma Pi sorority. The Art of Scientific Computing, 2nd ed. The Gamma distribution can be thought of as a sum of i. You again find yourself in a diving competition; in this competition, you take 3 dives, and each is scored by a judge from 0 to 1 1 being the best, 0 the worst. Plausible Reasoning in the 21st Century.
When you first took calculus, you probably learned a variety of different integration methods: Imagine that you were asked to calculate the following integral:. This does not look like a trivial integral to solve. Specifically, recall the Beta PDF from earlier in the chapter. Now that we have recognized what PDF this is similar to, we can use the fact that valid PDFs, when integrated over their support, must integrate to 1. Here, we are integrating from 0 to 1, which we know to be the support of a Beta.
Therefore and this is the big step we multiply and divide by the normalizing constant:. We are thus left with an elegant result:.