15A - Discrete random variables

Random variables

A random variable is a variable whose value cannot be predicted as the outcome is subject to variations due to chance. We can assign a numerical probability to each outcome of the random variable to produce a probability distribution.

Discrete random variables

A discrete random variable is a random variable where there is a finite, countable number of distinct events. For example:

• The number of Heads obtained when three coins are flipped.
• The number of people wearing shorts
• Rolling a '6' on a die

Continuous random variables

A continuous random variable is a random variable where there is an infinite number of events within a domain (set of values). For example:

• Height and weight of a person
• Volume, area or length of an object
• Temperature

Discrete random variables and their distributions

A discrete random variable is usually represented with a capital letter (..., X, Y, Z,).

Case study: Flipping three coins

Suppose we flip three fair coins. We can define a discrete random variable, X, as the number of Heads (H) obtained. Because this is a relatively simple experiment we can determine all possible outcomes and as the coin is fair we can also calculate their probabilities (a Tree diagram can also be useful in setting up a probability distribution):

The probability distribution for this discrete random variable would be:

Once we have set up a probability distribution for a discrete random variable we can use it to answer questions such as: what is the probability of flipping at least 1 Head?

Properties of discrete random variable distributions

For any discrete probability function, the following must be true:

• The value of p(x) is greater than or equal to zero and less than or equal to one:

15A - VIDEO EXAMPLE 1:

Suppose we through two fair 6-sided dice. Suppose we let the random variable, X, be the sum of the two dice.

• Use a suitable method to represent the sample space.
• Hence, determine the probability distribution for the discrete random variable, X.
• Find Pr(X > 8).

15A - VIDEO EXAMPLE 2:

Consider the discrete random variable X with the following probability distribution:

15A - VIDEO EXAMPLE 3:

Anna, Maddie and Grace try to go out to lunch every Friday. However, on any given Friday one or more of the girls may not be able to make it to lunch. The probability that Anna cannot make lunch is 0.2, that Maddie cannot make lunch is 0.4 and that Grace cannot make lunch is 0.15.

• If these are all independent events, determine the probability distribution for the discrete random variable, X, which measures how many of the girls can make it to lunch.
• The girls decide they will only go out for lunch if at least two of them are available. Find the probability that the girls will go out for lunch on any given Friday.
• What is the probability that all three girls went out for lunch, given that the group attended lunch (give you answer correct to 4 decimal places).