Examples

PMF of XY squared from chart

Suppose the joint PMF of and is given by this chart:

	0.2	0.2
	0.35	0.1
	0.05	0.1

Define .

(a) Find the PMF .

(b) Find the expectation .

Max and Min from joint PDF

Suppose the joint PDF of and is given by:

Find the PDFs:

(a)

(b)

Solution

(a)

(1) Compute CDF of :

Convert to event form:

Integrate PDF over the region, assuming :

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(2) Differentiate to find :

:

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(b)

(1) Compute CDF of :

Convert to event form:

Integrate PDF over the region:

Therefore:

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(2) Differentiate to find :

:

PDF of a quotient

Suppose the joint PDF of and is given by:

Find the PDF of .

Solution

(1) Find the CDF using logic:

Integrate over this region:

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(2) Differentiate to find PDF:

Compute :