W08 - Examples

Functions on two random variables

PMF of 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:
   • Interpret:
   • Integrate PDF over the region, assuming :
   • Insert PDF formula:
2. & Differentiate to find .
   • :

(b)

1. &&& Compute CDF of .
   • Convert to event form:
   • Consider complement event to interpret:
   • Integrate PDF over the region:
   • Compute integral:
   • Therefore:
2. & Differentiate to find .
   • :

PDF of a quotient

Suppose the joint PDF of and is given by:

Find the PDF of .

1. &&& Find the CDF using logic.
   • Convert to event form:
   • Re-express:
   • Diagram:
   • Compute:
2. & Differentiate to find PDF.
   • Compute :

Sums of random variables

Sum of parabolic random variables

Suppose is an RV with PDF given by:

Let be an independent copy of . So , but is independent of .

Find the PDF of .

Solution The graph of matches the graph of except (i) flipped in a vertical mirror, (ii) shifted by to the left.

When , the integrand is nonzero only for : When , the integrand is nonzero only for :

Final result is:

Discrete PMF formula for a sum

Prove the discrete formula for the PMF of a sum. (Apply the general formula for the PMF of .)

Vandermonde’s identity from the binomial sum rule

Show that this “Vandermonde identity” holds for positive integers :

Hint: The binomial sum rule is:

Set . Compute the PMF of the left side using convolution. Compute the PMF of the right side directly. Set these PMFs equal.

Convolution practice

• Suppose is an RV with density:
• Suppose is uniform on .

Find the PDF of . Sketch the graph of this PDF.

Exp plus Exp equals Erlang

Let us verify this formula by direct calculation:

Solution Let be independent RVs.

Therefore: Now compute the convolution:

This is the Erlang PDF:

Erlang induction step

By direct computation with PDFs and convolution, derive the formula:

Combining normals

Suppose , . Find the probability that .

Solution Define . Using the formulas above, we see , or for a standard normal . Then: