Consider an experiment where three coins are tossed and the number of heads is recorded. The experiment is called a random experiment, and the number of heads is a random variable.
A random variable allows you to use the language of functions when talking about probability.
A discrete random variable is a function from a discrete sample space to a set of numbers. A discrete random variable determines a set of probabilities so that the probability of a value occurring corresponds to the probability of occurrence of the corresponding event of the sample space.
Example Let the random variable X count the number of heads when tossing three coins. The sample space and values of X are listed in the table below.
Outcome | HHH | HHT | HTH | THH | TTT | TTH | THT | HTT |
Value | 3 | 2 | 2 | 2 | 0 | 1 | 1 | 1 |
The value that X takes is denoted by x.
To calculate the probabilities associated with the values of a random variable, just add the probabilites of elements of the sample space that produce the same value.
For the example above, \[ \begin{align} P(X=3) &= \frac{1}{8} \\ P(X=2) &= \frac{1}{8} + \frac{1}{8} + \frac{1}{8} \\ P(X=1) &= \frac{1}{8} + \frac{1}{8} + \frac{1}{8} \\ P(X=0) &= \frac{1}{8} \\ \end{align}\ \]
The probabilities associated with the values of a random variable are displayed in a probability distribution table. For the example above,
xi | 3 | 2 | 1 | 0 |
P(X=xi) | 0.125 | 0.375 | 0.375 | 0.125 |
The set of possible values of the random variable and their associated probabilities is called a probability mass function or a probability distribution function, denoted by f(x).
Add the probabilities in the probability row. Explain what this means.
What else can you say about the probabilities?
The graph of the probability mass function for the above distribution is
There are two constraints a probability distribution must satisfy:
Any probability \( P(X=x_i) \) is \( \ge \) 0.
Sum of probabilities = 1
These two constraints are really consequences of the definition of probabilty.
The probability distribution discussed above is called a discrete distribution because the random variable X takes on discrete values.
On the other hand, some random variables are not restricted to integer values but can vary over the real numbers. If a person's height is measured as 171 cm and the measuring instrument only measures in cms, then the person's actual height could be anything from 170.5 cm to 171.5 cm. It will, however, be a definite number - for example it might be 171.148456783482.
A continuous random variable is a random variable whose range is an interval on the real line. So the range of a random variable that records the height of a person in cms could be from 30 to 270.
This makes it impossible to write down all possible values for a continuous distribution in a table, as we can for a discrete random variable.
Find another example of a discrete distribution.
Find another example of a continuous distribution.
Continuous probability distributions will be investigated further in Year 12.
Guided Examples
O E Qs
Uniform Distribution
Consider the probability of any of the numbers 1 to 6 appearing on the top face of a dice when it is thrown. Each of the numbers 1 to 6 is equally likely. So from the definition of probability, the probability distribution table for this is
xi | 1 | 2 | 3 | 4 | 5 | 6 |
P(X=xi) | 0.166 | 0.166 | 0.166 | 0.166 | 0.166 | 0.166 |
And you can confirm that each individual probability is non-negative and that the probabilities sum to one. Because the probabilities are equal for each outcome, this is called a uniform distribution.
The graph of the probability mass function for the uniform distribution is
What can you say about the type of event associated with a uniform distribution?
Binomial Distribution
The probability distribution for the introductory example of tossing three coins is
xi | 3 | 2 | 1 | 0 |
P(X=xi) | 0.125 | 0.375 | 0.375 | 0.125 |
This can be regarded as tossing one coin three times. The possible outcome on each toss is one of two - heads or tails. The probability for a particular event, say HHT, is the product of the probabilities on each toss: \[ P(HHT) = (\frac{1}{2})(\frac{1}{2})(\frac{1}{2}) \ \]
Because there are only two possible outcomes at each toss, this is called a binomial ditribution.
Calculate the probability of obtaining two heads, in any order, and compare your result with the table above.
There is a shorthand for specifiying sums of terms which will be used in the next few lessons. The greek letter (capital) sigma \[ \sum \ \] is used to indicate that the expression following it should be summed. The range of summation is specified by an index attached to sigma. Some simple examples are: \[ \begin{align} \sum_{i=1}^n 1 = 1 + 1 + ... + 1 = n \quad \end{align}\ \] and \[ \sum_{i=1}^n i = 1 + 2 + 3 + ... + n \ \] Normally the summation index in the expression is attached to other variables: \[ \sum_{i=1}^n f_i x_i = f_1 x_1 + f_2 x_2 + ... + f_n x_n \ \]
As an immediate application, the property that the probabilities of a discrete random variable X sum to 1 is expressed as \[ \sum_{i=1}^n P(X = x_i) = 1 \quad \ \]