A solid grasp of probability distributions of discrete random variables is essential. This detailed exploration covers the creation of probability distribution tables, the calculation of expected value $E(X)$ and variance $Var(X)$, accompanied by practical examples. These concepts are pivotal in understanding statistical analysis and data interpretation.

Probability Distributions

• A probability distribution describes how likely different outcomes are in an experiment.
• For discrete random variables, outcomes are distinct, like countable numbers.

Discrete Random Variables

• These variables take specific values, often whole numbers.
• Examples: Number of correct answers on a test, number of heads in coin flips.

Probability Distribution Tables

• These tables show probabilities for each outcome of a discrete random variable.

Example: Coin Toss

• Experiment: Tossing a fair coin twice.
• Random variable X = number of heads.
• Possible X values: 0, 1, 2.
• Each outcome (head or tail) is equally likely.

Expected Value (E(X))

• Represents the 'average' outcome of a random variable.
• Calculated as: E(X) = Sum of [x * P(x)].
• Example: Coin Toss, E(X) = 0 0.25 + 1 0.50 + 2 * 0.25 = 1.
• In two coin tosses, expect 1 head on average.

Variance (Var(X))

• Measures how spread out the data is.
• Calculated as: $Var(X) = E (X - E(X))^2$.
• Example: Coin Toss, $Var(X) = 0.25 (0 - 1)^2 + 0.50 (1 - 1)^2 + 0.25 * (2 - 1)^2 = 0.5$.
• Variance for number of heads in two tosses is 0.5.