Random Variables

A Random Variable is neither Random nor a Variable.

Gian-Carlo Rota

A Sample Space, $\QTR{Large}{S}$ , is a set of all possible outcomes of some random experiment.
A Random Variable, is function, usually numerical, function on a Sample Space. We will also use capital letters, such as $\QTR{Large}{X}$ , to denote Random Variables.
The probability that $\QTR{Large}{X}$ returns the value $\QTR{Large}{x}$ , is $\QTR{Large}{p}$ is written,
We also talk about Discrete Random Variables is one which may take on at most countable number of distinct values and Finite Random Variables which can take only a finite number of distinct values. By definition, Finite Random Variables are also Discrete.

Flip a coin - Informally, Possible Outcomes {H,T} each with probability $\QTR{Large}{1/2.}$

More formally, $\QTR{Large}{S=}$ {H,T} and $\QTR{Large}{X}$ is a Finite Random Variable taking on the values 0 and 1 , for H and T , $\$ finally

01 $\QTR{Large}{)=1/2}$ .
Flip a coin twice - Possible Outcomes {HH,HT,TH,TT} each with probability $\QTR{Large}{1/4.}$

What would be an appropriate $\QTR{Large}{X}$ ?
Learn the content of a byte in memory - Possible Outcomes {[00000000],...[01001110],... } each with probability
Given that we know that the content of a byte contains the ASCII code for a lower case letter of the english alphabet.

What is the probability that the byte contains an "e"?

Knowing nothing about a deck of cards that is presented a person is asked to:

Choose a card from a standard deck - Possible Outcomes {AS,2S,.....,KS,AH.........KC} each with probability $\QTR{Large}{1/52}$
1. What is the Suit ? - Possible Outcomes {S,H,D,C} each a set of 13 cards, each with probabililty $\QTR{Large}{1/4}$
2. What is the Rank? - Possible Outcomes {A,2,3,4,5,6,7,8,9,10,J,Q,K} each a set of 4 cards and each with probabililty $\QTR{Large}{1/13}$
Choose from two standard decks that have been mixed together

- Possible Outcomes {AS,2S,.....,KS,AH.........KC} each with probabililty $\QTR{Large}{1/52}$
1. What is the Suit ? - Possible Outcomes {S,H,D,C} each with probablilty $\QTR{Large}{1/4}$
2. What is the Rank?- Possible Outcomes {A,2,3,4,5,6,7,8,9,10,J,Q,K} each with probabililty $\QTR{Large}{1/13}$

Choose a card from a standard deck that has two identical Jokers added

-Possible Outcomes {AS,2S,.....,KS,AH.........KC,Jo} each standard card with probabililty $\QTR{Large}{1/54}$ and Jo with probabililty $\QTR{Large}{2/54}$
1. What is the Suit ??- Possible Outcomes { $\QTR{Large}{?}$ } each with probablilty $\QTR{Large}{?}$
2. What is the Rank? - Possible Outcomes { $\QTR{Large}{?}$ } each with probablilty $\QTR{Large}{?}$
Choose a card from a standard deck with AH and 3C removed

-Possible Outcomes {AS,2S,.....,KS,AH.........KC } each standard card (not AH or 3C ) with probabililty $\QTR{Large}{1/50}$
1. What is the Suit? - Possible Outcomes {S,H,D,C} with probablilties $\QTR{Large}{?}$
2. What is the Rank? - Possible Outcomes {A,2,3,4,5,6,7,8,9,10,J,Q,K} with probablilties $\QTR{Large}{?}$
3. Given that you know you drew a Heart, What are the probabilities for the various ranks?
4. Give that you know you drew a 3, What are the probabilites for the various suits?