25
Dec 21

Analysis of problems with conditioning

category: AP Stats and Business Stats No Comments

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: Y is conditioned on X.

Your life will be easier if you follow the notation used in the guide: use p for probability mass functions (discrete variables) and f for (probability) density functions (continuous variables).

a) If Y|X and X both are discrete (Example 5.1, Example 5.13, Example 5.18):

p_{Y}\left( y\right) =\sum_{Set}p_{Y\vert X}\left( y\vert x\right) p_{X}\left( x\right) .

b) If Y|X and X both are continuous (Activity 5.6):

f_{Y}\left( y\right) =\int_{Set}f_{Y\vert X}\left( y\vert x\right) f_{X}\left( x\right) dx.

c) If Y|X is discrete, X is continuous (Example 5.2, Activity 5.5):

p_{Y}\left( y\right) =\int_{Set}p_{Y\vert X}\left( y\vert x\right) f_{X}\left( x\right) dx

d) If Y|X is continuous, X is discrete (Activity 5.12):

f_{Y}\left( y\right) =\sum_{Set}f_{Y\vert X}\left( y\vert x\right) p_{X}\left( x\right) .

In all cases you need to figure out Set 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 X denote the number of hurricanes which form in a given year, and let Y denote the number of these which make landfall. Suppose each hurricane has a probability of \pi making landfall independent of other hurricanes. Given the number of hurricanes x, then Y can be thought of as the number of successes in x independent and identically distributed Bernoulli trials. We can write this as Y|X=x\sim Bin(x,\pi ). Suppose we also have that X\sim Pois(\lambda ). Find the distribution of Y (noting that X\geq Y ).

Solution

Step 1. The number of hurricanes X takes values 0,1,2,... and is distributed as Poisson. The number of landfalls for a given X=x is binomial with values y=0,...,x. It follows that Set=\{x:x\ge y\}.

Write the general formula for conditional probability:

p_{Y}\left( y\right) =\sum_{x=y}^{\infty }p_{Y\vert X}\left( y\vert x\right) p_{X}\left( x\right) .

Step 2. Specifying the distributions:

p_{X}\left( x\right) =e^{-\mu }\frac{\mu ^{x}}{x!}, where x=0,1,2,...,

and

P\left( Bin\left( x,\pi \right) =y\right) =p_{Y\vert X}\left( y\vert x\right) =C_{x}^{y}\pi ^{y}\left( 1-\pi \right) ^{x-y} where y\leq x.

Step 3. Reduce the result to one of known distributions:

p_{Y}\left( y\right) =\sum_{x=y}^{\infty }C_{x}^{y}\pi ^{y}\left( 1-\pi \right) ^{x-y}e^{-\mu }\frac{\mu ^{x}}{x!}

(pull out of summation everything that does not depend on summation variable
x)

=\frac{e^{-\mu }\mu ^{y}}{y!}\pi ^{y}\sum_{x=y}^{\infty }\frac{1}{\left( x-y\right) !}\left( \mu \left( 1-\pi \right) \right) ^{x-y}

(replace x-y=z to better see the structure)

=\frac{e^{-\mu }\mu ^{y}}{y!}\pi ^{y}\sum_{z=0}^{\infty }\frac{1}{z!}\left( \mu \left( 1-\pi \right) \right) ^{z}

(using the completeness axiom \sum_{x=0}^{\infty }\frac{\mu ^{x}}{x!}=e^{\mu } for the Poisson variable)

=\frac{e^{-\mu }}{y!}\left( \mu \pi \right) ^{y}e^{\mu \left( 1-\pi \right) }=\frac{e^{-\mu \pi }}{y!}\left( \mu \pi \right) ^{y}=p_{Pois(\mu \pi )}\left( y\right) .

 

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