Conditional Expectation

Definition (Conditional Expectation).

Let the joint pmf of be the conditional expectation of given

Example

Important

Refer 7.5 Conditional Expectaion from A first course in Probability by - Sheldon Ross (page number: 313) Condtional Expectation Condtional Expectation
Example 5a 5c 5e 5g 5h 5i which were done in class but not may not be completely written in this notes. Don't worry they are about the same

, are independent independent and identical identicalBin( , )

this distribution is similar to hyper-geometry with total balls and black balls and ball are picked from it

this has the expectation

Remark.

Let be hyper-geometry with total balls and black balls and ball are picked from it where picking black is success then

Definition.

is the function of Random veriable whose value at is

Remark.

is a Random variable depends on Y

Remark.

If , independent then

Theorem.

Proof.

Example

Miner trapped in mine
There are 3 roads that miner can escape to each road has an equal probability to be chose and the fist road takes 5 hours to and he comes back to the same place and the middle road 3 hours and to escape and last road comes back to the same place after 7 hours

Let be the RV time taken to escape is the door number

independent independent RV with for all
take value
is independent of all
what is

Exercise.

Roll a pair of dice if the sum is 2,3 or 12 player loses if 7 or 11 wins if sum is any other then the player continues to roll the dice sum is either 7 or if it is 7 player losses
if it is player wins
is a RV rolls in a game