Discrete Random Variables: PMF, CDF, and Expectation

• This lesson covers discrete random variables, probability mass functions (PMF), cumulative distribution functions (CDF), and expectation.

Challenge - Decision Making

• A game stall involves picking balls from a bag containing 3 red (3R) and 2 blue (2B) balls, with the pick replaced each time, repeated 4 times.
• If four red balls (RRRR) are picked, the player wins +150₹; otherwise, they lose -10₹.
• The questions to consider are: Would you play this game? At what reward would you be willing to play if not at the current setup?
• Doing this activity and discussing the probability can be valuable

Recap - Experiment, Sample Space (SS), and Events

• Tossing a fair coin: The sample space (SS) is {H, T}, with events being getting heads or tails.
• Tossing a fair coin twice: The SS is {HH, HT, TH, TT}, with events including at least one head, no heads, both heads, or the same result on both tosses.
• Rolling a fair die: The SS is {1, 2, 3, 4, 5, 6}, and events can be getting a specific number (e.g., 6) or an even number.
• Rolling two dice: The SS has 36 possibilities. Events can be getting a sum k or having the same number on both dice, or one of them being a 6.

From Events to Random Variables

• A random variable (X) assigns numerical values to outcomes of events.
• For a fair coin toss, if X = number of heads, then for outcome H, X=1, and for outcome T, X=0.
• When tossing a fair coin twice, if X represents outcomes where both tosses result in the same face, then for HH and TT, X=1, and for HT and TH, X=0.
• When rolling a fair die, let X be 1 if the result is 6 (got 6) and 0 otherwise (not 6).

Event example

• RV is "Same number on two dice"
• Tabulating possible values reveals RV=1 for outcomes like (1,1), (2,2).
• RV=0 for unlike outcomes such as (1,2), all other cases.
• Outcomes of rolling 2 dice: (1,1), (1,2), ..., (6,6).
• The RV of the number of sixes can also be calculated
• The RV of the sum of numbers on the die can also be calculated

Why Random Variables?

• E represents a subset of outcomes
• RV assignes Numbers to the outcomes
• Die Roll E: Getting Even where X=0 = {1,3,5} and X=1 = {2,4,6}
• The use of random variables is to define specific outcomes descriptively or to enable analytical and computational mathematical modeling.
• Using 'X = Number of Red Balls' is more informative than 'X = 4 Red Balls or Not'

Discrete Random Variable (DRV)

• DRV can take on a countable number of distinct values.
• Each DRV has an associated probability, and the sum of all probabilities equals 1.
• Examples include the number of heads when flipping k coins, the number of 1s when rolling k dice, and the number of sick patients arriving in a hospital in an hour.

Probability Mass Function (PMF)

• The PMF function describes the probability of a discrete random variable taking a specific value.
• PMF maps each possible value of a random variable to its corresponding probability of occurrence.

Probability Mass function application

• For a game of drawing balls, to find what's the probability of getting 4 Red balls
• P(X=0), P(X=1), P(X=2), and P(X=3) must be calculated.

PMF - Function Describing Probability

• PMF describes the probability of a discrete random variable taking a specific value.
• PMF maps each possible value of a random variable to its corresponding probability of occurrence.
• Flipping a coin: Equally likely outcomes of obtaining heads
• Rolling a die: Each side is equally likes: 1 to 6
• Tossing 3 coins: The probability is not equally likely, some sides will occur more often

Experiments as RVs

• Roll two dice twice
• Consider the Sum of values of the two dice
• All outcomes are not equal likley
• The PMF can be generated to calculate probabilities, e.g, P(X=10) for a roll is ~0.09

Cumulative Distribution Function (CDF)

• CDF gives the probability that a random variable is less than or equal to a certain value.
• Use it to find the probabilities for range of values efficiently
• Such as the probability of winng when rolling a die, where probabilities can be calculated as the sum is between 6 and 10
• Rolling a die two times: X is the Sum of values on both throws
• Finding probabilies requires the need to add some of the PMF
• P(6 ≤ X ≤ 10) = = 0.13 +0.17 +0.14 + 0.11 +0.09
• Using CDF provides the same answer more efficiently
• P(6 ≤ X ≤ 10) = F(10) - F(5)

Summary - Discrete RV, PMF, CDF

• Random Variable: A numerical representation of events for analysis.
• Discrete Random Variable (DRV): A random variable with a countable set of distinct values.
• Probability Mass Function (PMF): A function P(X), which assigns probabilities to each DRV value.
• Cumulative Distribution Function (CDF): A function F(x) = P(X ≤ x) giving the probability of values ≤ x.

Expectation, Expected Value, Decision Making

• Expected values are crucial in decision-making processes as they help in calculating and comparing the expected outcomes of different actions.
• Expectation and Expected Value can calculate expected earning
• P(Win) is 0.13 (very low) but stakes are also high
• In the example from earlier slides, it can be showed that ~2.42 red balls will appear from trials
• Lets define X as amount earned/lost in one play
• For example calculating expected earning over multiple trials can be calculated, where the earnings are >0, and it may be okay to play the game