Examples

Conditional PMF, fixed event, expectation

Suppose measures the lengths of some items and has the following PMF:

Let be the event that .

(a) Find the conditional PMF of given that is known.

(b) Find the conditional expected value and variance of given .

Solution

(a)

Conditional PMF formula with plugged in:

Compute by adding cases:

Divide nonzero PMF entries by :

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(b)

Find :

Find :

Find using “short form” with conditioning:

Conditional PMF, variable event, via joint density

Suppose and have joint PMF given by:

Find and .

Solution

Marginal PMFs:

Assuming or , for each we have:

Assuming , , or , for each we have:

Proof of Iterated Expectation, continuous case

Prove Iterated Expectation for the continuous case.

Conditional expectations from joint density

Suppose and are random variables with joint density given by:

Find . Use this to compute .

Solution

(1) Derive the marginal density :

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(2) Use to compute :

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(3) Use to calculate expectation conditioned on the variable event:

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(4) Apply Iterated Expectation:

Set . By Iterated Expectation, we know that . Therefore:

Notice that , so , and Iterated Expectation says that .

Flip coin, choose RV

Suppose and represent two biased coins, giving 1 for heads and 0 for tails.

Here is the experiment:

1. Flip a fair coin.
2. If heads, flip the coin; if tails, flip the coin.
3. Record the outcome as .

What is ?

Solution

Let describe the fair coin. Then:

Sum of random number of RVs

Let denote the number of customers that enter a store on a given day.

Let denote the amount spent by the customer.

Assume that and 8i$.

What is the expected total spend of all customers in a day?

Solution

A formula for the total spend is .

By Iterated Expectation, we know .

Now compute as a function of :

Therefore and and .

Then by Iterated Expectation, 400$.