Basic Concepts in Probability and Statistics

Abstract

Exercise 1.1 (Empirical Probabilities). The relative frequency of an event is the fraction of times the event occurs divided by the total number of experiments. Suppose a coin is tossed 200 times. The outcome of 100 consecutive pairs of tosses is summarized in the contingency table 1.1, where H j and T j indicates the number of heads and tails on the jth toss, respectively. For instance, the "21" listed in the top right entry of the table means that 21 out of 100 pairs was "heads-tails," and the empirical probability of "heads-tails" is 21/100. What is the empirical probability of heads on the first toss of a pair, denoted P (H 1)? What is the empirical probability of a heads in the second toss of a pair, denoted P (H 2)? What is the empirical probability of the joint event H 1 and T 2 , denoted P (H 1 , T 2)? What is the empirical probability of T 2 given H 1 , denoted P (T 2 |H 1)? Verify that the empirical conditional probability is the ratio of the joint probability over the marginal: P (T 2 |H 1) = P (T 2 H 1)/P (H 1). (Incidentally, this exercise suggests why unconditional probabilities are called marginal probabilities, namely because they can be calculated by summing values in a table along rows and columns and writing the results along the margins of the table.) Table 1.1 Sample Outcome of 100 pairs of coin tosses H2 T2 H1 30 21 T1 22 27 Exercises for Statistical Methods for Climate Scientists.