Section 4.2: Random Variables

  1. Denote a random variable by a capital letter and a particular value the random variable can take by a lowercase letter. For example, we might denote the number appearing after the roll of a six-sided die by \(X\) and one of the particular numbers by \(x\).
  2. Explain in what sense a random variable is a number that could have been otherwise.
  3. Explain what a probability distribution tells us about a random variable.
  4. Denote the probability distribution of a random variable \(X\) by \(P(X = x)\), where \(x\) is a particular value the random variable can take.
  5. Define what it means for a random variable to be discrete or continuous, and distinguish between discrete and continuous random variables when given their description.
  6. Construct a probability histogram for a discrete random variable given its probability distribution.
  7. Specify the properties that a probability distribution must satisfy, and check if a given probability distribution satisfies those properties.
  8. Explain what the expectation of a random variable means under a frequency-based model of probability.

Section 4.3: Binomial Probability Distributions

  1. Specify the properties that must be true of a procedure for its outcome to be well-modeled by a binomial random variable.
  2. Identify the roles played by \(n\), \(p\), and \(x\) in the probability distribution of a binomial random variable.
  3. Extract \(n\), \(p\), and \(x\) from the description of a procedure whose outcome is well-modeled by a binomial random variable.
  4. Interpret and use a table like Table A–1 from Appendix A of Triola and Triola to find \(P(X = x; n, p)\).
  5. Use (but do not memorize) the formula \(P(X = x; n, p) = \dfrac{n!}{(n-x)!x!} p^{x} (1 - p)^{n - x}\) to find the probability of \(x\) successes in \(n\) trials.
  6. Convert events such as “at least 1,” “at most 2,” “fewer than 3,” and “more than 10” to the appropriate values of a binomial random variable with \(n\) trials.
  7. Compute the probability of events like those in the previous learning objective using the probability distribution of a binomial random variable.