Section 3.1: Random Variables

  1. Give the definition of a random variable as a function from a sample space to the real numbers.
  2. Properly use the convention of denoting a random variable by an uppercase Roman letter (e.g. \(X, Y, Z\)), a particular value that random variable can take by the corresponding lowercase Roman letter (e.g. \(x, y, z\)), and its range by the corresponding calligraphic Roman letter (e.g. \(\mathcal{X}, \mathcal{Y}, \mathcal{Z}\)).
  3. Distinguish between a discrete random variable and a continuous random variable.
  4. Explain in what sense a random variable is a ‘‘number that could have been otherwise.’’
  5. Given a sample space and a description of a random variable, determine the range of the random variable, and determine what values of the sample space map to each value of the random variable’s range.

Section 3.2: Probability Distributions for Discrete Random Variables

  1. Define the probability mass function \(p(x) = P(X = x)\) for a random variable \(X\) in terms of the underlying sample space \(\mathcal{S}\).
  2. State the two characteristics a function \(p\) must have to be a valid probability mass function, and identify when a given function \(p\) does or does not satisfy these conditions.
  3. Use a probability mass function \(p\) to compute the probability \(P(X \in Q)\) that \(X\) falls in some query set \(Q \subseteq \mathbb{R}\).
  4. Use the “Probability Dictionary” to map from statements such as “at least \(x\),” “at most \(x\),” etc., to the corresponding query set \(Q\).
  5. Define the cumulative distribution function \(F(x) = P(X \leq x)\) for a random variable \(X\) in terms of the underlying sample space \(\mathcal{S}\).
  6. State the four properties a function \(F\) must have to be a valid cumulative distribution function.
  7. Define what it means for a function \(f\) to be continuous and right-continuous.
  8. Sketch the graph of a cumulative distribution function \(F\) for a discrete random variable \(X\) given either its \(p\) or \(F\).
  9. Compute \(F\) from \(p\), and vice versa.
  10. Compute the probability \(P(a < X \leq b)\) of a standard query using \(F\).
  11. Compute the probability \(P(a \square X \square b)\) of a non-standard query using \(F\).