## 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.

### General scheme

**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

axiom.

### Example 5.1

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 ).

### Solution

**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:

where

and

where

**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)