Random Variables

1. Random Variables November 23, 2009

2. Discrete Random Variables • A random variable is a variable whose value is a numerical outcome of a random phenomenon. • When each value of a random variable can be assigned a probability, the random variable is discrete.

3. Probability Distributions • This list of probabilities assigned to each possible value of a random variable X is called the probability distribution of X. • The probability distribution can be written as a table, or as a histogram (called a probability histogram). • In order to be a legitimate probability distribution, the probabilities must fall between 0 and 1 and sum to 1.

4. Example 1 (Uniform Distribution) • Imagine picking a digit from the (infinite) decimal expansion for . Let X be the random variable whose value is the digit you pick. The probability for each digit is equally likely. • Find the probability distribution and make a probability histogram. What is P(X > 3)?

5. Example 2 • Spell-checking catches “nonword errors,” which result in a string of letters that is not a word (such as “teh” for “the”). • Let X represent the number of nonword errors in a 250 word essay. • The variable X has the following probability distribution:

6. Example 2 Continued • Verify that this gives a legitimate probability distribution. • Write the event “at least one nonword error” in terms of X. What is the probability of this event? • Describe the event X ≤ 2 in words. What is the probability that X < 2?

7. Assignment • Page 461: 7.2,7.3 and 7.4 • Page 475: 7.7 and 7.8 • Page 477: 7.12,7.14,7.15 and 7.20 • Due Wednesday