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W11 - Examples

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Conditional distribution

Conditioning on a fixed event

Suppose measures the lengths of some items and has the following PMF: Let be the event that .

(a) Find the PMF of conditioned on .

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

Solution (a)

  1. By the definition:
  2. Adding exclusive probabilities:
  3. Note that . Therefore:

(b)

  1. Find :
  2. Find :
  3. Find :

Conditioning on variable events, discrete PMF function

Suppose and have joint PMF given by:

Find and .

Solution First compute the marginal PMFs:

Therefore, assuming or , then for any we have:

And, assuming , , or , then for any we have:

Conditional expectation

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 First derive the marginal density :

Use to compute :

Use to calculate expectation conditioned on the variable event:

So, 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$.