Chapter 9

The idea of probability

1. Explain why we need to talk about chance when we consider a a sample from a simple random sample sampling design.
2. Define probability in terms of a proportion related to long-run frequencies.

Probability rules

1. State the range of values that a probability can take.
2. Recognize when a number can and cannot be a probability.
3. Relate the probability that an event occurs to the probability that the event does not occur.
4. Characterize the probability of an event between 0 and 1 in terms of whether the event will:
   • always occur
   • never occur
   • sometimes occur

Random variables

1. Explain why some variables are considered “random” in statistics, and explain in what sense a random variable is a “number that could have been otherwise.”
2. Give examples of random variables that result from simple random sampling.
3. Recognize and use the notation of an upper case letter (like \(X, Y,\) or \(Z\)) for a random variable, and a lower case letter (like \(x, y,\) or \(z\)) for a particular value the random variable could take.
4. Given the definition of a random variable, determine whether the random variable is discrete or continuous.

Discrete versus continuous probability models

1. State the properties of a probability distribution for a discrete random variable, and identify whether a given table can or cannot be a probability distribution.
2. Use a probability distribution to answer probability queries such as \(P(X > a)\), \(P(a < X < b)\), \(P(X \leq a)\), etc., for a discrete random variable \(X\).
3. State the properties of a density curve for a continuous random variable.
4. Relate the probability that a continuous random variable \(X\) falls in an interval to its probability curve.
5. Given a rectangular or triangular density curve for a random variable \(X\), find the probability that \(X\) falls in a given interval.