By Dauxois J.-Y., Druihlet P., Pommeret D.

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**Extra info for A Bayesian Choice Between Poisson, Binomial and Negative Binomial Models**

**Example text**

And" means multiply. Of course, we warned you that these rules don't hold in complete generality. Let's explore these rules a bit. In what follows, let X be the number of dots on the top face of a fair die: (a) Under what conditions do these simple rules hold? (b) Which simple rule calculates P(2 :S X :S 5)? Explain. 3 On one draw of a card from a well-shuffled deck of 52 playing cards, what's the probability that you do NOT draw an eight? Do this (a) directl y, by just counting, (b) using one of our three probability rules.

Here are the dice: • all faces are equally likely; • two dots come up 50% of the time and the other faces are equally likely; • two dots come up 40% of the time, five dots 20% of the time, and the other faces are equally likely. (a) Compare the predictability of your gain/loss for each of the three dice. (b) In part (a), if the player breaks even in the long run, the gambling house won't be a profitable business! Suppose the house wants to make an average profit of 50 cents per play; what should they charge to play?

Clearly, the expected value by itself will not suffice. As we see in the loaded die above, the expected value misses an important characteristic of the model for this die. It misses the fact that the probability is "spread" to the extreme values of X. We need a parameter to capture this characteristic of the model, the "spread" or "dispersion" of the values from the mean. 5=/-tx We'll discuss graphical presentations of distributions in more detail later, but as you can see, the graph for the loaded die shows at a glance that most of the probability for that die is concentrated at the extreme values of X.

### A Bayesian Choice Between Poisson, Binomial and Negative Binomial Models by Dauxois J.-Y., Druihlet P., Pommeret D.

by Joseph

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