# Random Variables November 23, 2009 ### Random Variables

November 23, 2009

### 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

.

### 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.

### 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)?

### 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: errors 0 Prob. 0.1

1 0.2

2 0.3

3 0.3

4 0.1

### 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?

### 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