6.2. Discrete Random Variables

A discrete random variable can take countable number of distinct values. Examples: Tossing a coin, rolling a die, number of defective items in a batch, etc.

A discrete random variable is defined as one that can take on a finite or countably infinite set of values. This means that we can list the possible outcomes without needing to resort to fractions or decimals. For example, when tossing a coin, the outcomes are either Heads (H) or Tails (T). Similarly, when rolling a die, the distinct values can be 1, 2, 3, 4, 5, or 6. Other examples include the count of defective items in a manufacturing process, which can be 0, 1, 2, etc.

Imagine a teacher counting the number of students who pass an exam. The possible outcomes can only be whole numbers: 0 students pass, 1 student passes, 2 students pass, and so on. This counting nature reflects the essence of discrete random variables.

The PMF of a discrete random variable X is defined as: 𝑃(𝑋 = 𝑥 ) = 𝑝 where ∑𝑝 = 1 and 0 ≤ 𝑝 ≤ 1. Example: If X is the number on a fair six-sided die: 𝑃(𝑋 = 𝑥) = 1/6, 𝑥 = 1,2,3,4,5,6.

The Probability Mass Function (PMF) is a function that gives the probability of each outcome for a discrete random variable. Each value x that the variable can take corresponds to a probability p. The sum of all probabilities for all possible outcomes must equal 1, reflecting the certainty that one of the outcomes will occur. For example, when rolling a fair die, each side has a probability of 1/6 because there are 6 equal possible outcomes.

Consider rolling a six-sided die during a game. You know that each outcome (1, 2, 3, 4, 5, and 6) has the same chance, which is 1 in 6. This consistent chance for each outcome illustrates how the PMF assigns probabilities to discrete outcomes.

The CDF of a discrete random variable X is: 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∑𝑃(𝑆𝑃𝑥𝑖 ) where 𝑖, 𝑥 ≤ 𝑥.

The Cumulative Distribution Function (CDF) is a function that describes the probability that a discrete random variable X takes on a value less than or equal to x. It is calculated by summing the probabilities obtained from the PMF for all values that are less than or equal to x. This cumulative aspect allows us to see the total probability up to a certain value.

Think of the CDF like a score tally in a game. If you were to count all the players who scored 3 points or less, you would be adding up their probabilities. If you know some players scored 0, others scored 1, and a few scored 2, the CDF helps you accumulate that total probability to give you insights into how many players scored at or below a certain point.

𝐸(𝑋) = ∑𝑥 𝑃(𝑋 = 𝑥 ).

Expectation, often referred to as the mean, is a measure of the center of a probability distribution for a discrete random variable. It is calculated by taking each possible value of the random variable, multiplying it by its probability, and then summing these products. This yields a single number that represents the average outcome you can expect.

Imagine a scenario where you're playing a lottery where you can win $0,$10, or $20 with probabilities of 0.5, 0.3, and 0.2, respectively. To find the expectation, you calculate: (0 * 0.5) + (10 * 0.3) + (20 * 0.2) =$6. This means that on average, you can expect to win $6 per lottery ticket if you played many times.

Var(𝑋) = 𝐸[(𝑋− 𝜇)²] = ∑(𝑥 − 𝜇)²𝑃(𝑋 = 𝑥 ).

Variance measures the spread of a set of values from their mean. It is calculated by taking the average of the squared differences from the mean (𝜇). By summing these squared differences, weighted by their probabilities, variance provides insight into how much variability there is in the outcomes of the random variable. A small variance indicates outcomes are close to the mean, while a large variance shows they are more spread out.

Think of variance like evaluating the scores of students in a class. If all students scored close to the average score, the variance is low, indicating consistency. If some scored very high and some very low, the variance is high, demonstrating a wide range of performances. This concept helps us understand the variability within the data.

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