In your own words, provide an interpretation of the mean (or expected value) of a discrete random variable.

Verified step by step guidance

1. Start by understanding that the mean or expected value of a discrete random variable represents the long-run average outcome if the random experiment were repeated many times.
2. Recognize that each possible value of the random variable is weighted by its probability of occurrence, reflecting how likely each outcome is.
3. Mathematically, the expected value \(E(X)\) is calculated by summing the products of each value \(x_i\) and its corresponding probability \(P(X = x_i)\), expressed as: \[E(X) = \sum_i x_i \cdot P(X = x_i)\]
4. Interpret this formula as a weighted average, where more probable outcomes have a greater influence on the expected value.
5. Conclude that the expected value provides a single summary measure that predicts the average result you would expect over many trials of the random process.

Key Concepts

Discrete Random Variable

A discrete random variable is a variable that can take on a countable number of distinct values, each with an associated probability. Examples include the number of heads in coin tosses or the roll of a die. Understanding this helps frame what outcomes the mean summarizes.

Mean (Expected Value)

The mean or expected value of a discrete random variable is the weighted average of all possible values, where each value is multiplied by its probability. It represents the long-run average outcome if the experiment is repeated many times.

Interpretation of Expected Value

Interpreting the expected value involves understanding it as a measure of central tendency or the 'center of mass' of the probability distribution. It provides insight into the typical or average outcome one can expect from the random variable.