Analysis of problems with conditioning
These problems are among the most difficult. It's important to work out a general approach to such problems. All references are to J. Abdey, Advanced statistics: distribution theory, ST2133, University of London, 2021.
Step 1. Conditioning is usually suggested by the problem statement: is conditioned on .
Your life will be easier if you follow the notation used in the guide: use for probability mass functions (discrete variables) and for (probability) density functions (continuous variables).
a) If and both are discrete (Example 5.1, Example 5.13, Example 5.18):
b) If and both are continuous (Activity 5.6):
c) If is discrete, is continuous (Example 5.2, Activity 5.5):
d) If is continuous, is discrete (Activity 5.12):
In all cases you need to figure out over which to sum or integrate.
Step 2. Write out the conditional densities/probabilities with the same arguments
as in your conditional equation.
Step 3. Reduce the result to one of known distributions using the completeness
Let denote the number of hurricanes which form in a given year, and let denote the number of these which make landfall. Suppose each hurricane has a probability of making landfall independent of other hurricanes. Given the number of hurricanes , then can be thought of as the number of successes in independent and identically distributed Bernoulli trials. We can write this as . Suppose we also have that . Find the distribution of (noting that ).
Step 1. The number of hurricanes takes values and is distributed as Poisson. The number of landfalls for a given is binomial with values . It follows that .
Write the general formula for conditional probability:
Step 2. Specifying the distributions:
Step 3. Reduce the result to one of known distributions:
(pull out of summation everything that does not depend on summation variable
(replace to better see the structure)
(using the completeness axiom for the Poisson variable)
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