III. Probability
B. Discrete Probability Distributions
In this section
 Valid Discrete Probability Distributions
 Mean of a Discrete Probability Distribution (Expected Value)
 Standard Deviation of a Discrete Probability Distribution
1.
Valid Discrete Probability Distributions
First it is important to make sure that we distinguish between a discrete and continuous
random variable. While the idea of a probability distribution is the same in both cases, the
way we have to look at the probability distribution is different. We will focus on the idea
of probability for a discrete random variable here.
Discrete Random Variable – potential values are separated points on the number line
(finite or countable number of values)
Continuous Random Variable – potential values fall in an interval on the number line
(infinite number of values)
A discrete random variable will include a finite or countable number of outcomes.
Therefore, we assign probabilities to each value of the random variable.
Probability distribution – the collection of
P (x)
values for all possible x values
For a discrete probability distribution to be valid, it must satisfy the following:
1. The variable must be discrete
2. 0  P ( x )  1 for all x
3.
 Px   1
Example 1
Flip a coin twice
x = number of tails
Identify the probability distribution for
x
2.
Mean of a Discrete Probability Distribution (Expected
Value)
Expected Value – the mean of a probability distribution (This value represents the same
quantity as the mean did in section II. B. The difference in the calculation comes from the
fact that outcomes in a discrete probability distribution are not equally likely in most
cases. Thus, this calculation is a weighted average. If all probabilities are the same then
the calculation from section II. B. will give the same outcome. The other big difference
here as compared to what we looked at in section II. B. is that we are now dealing with a
population mean and not a sample mean. In a probability distribution, we have all of the
possible outcomes meaning that we are describing the population.)
The population mean for a discrete probability distribution is:
   xP(x)

is a lowercase Greek letter said as ‘mu’ (Can be any real number)
The difference in notation for the mean and standard deviation as compared to section
II.B. is that now we are dealing with population values. It is very important to remember
the notation for the sample and population because later in the course you will see both
population and sample values in the same formula.
Example 2
Roll a die and X is the number rolled. Find the mean of X.
3.
Standard Deviation of a Discrete Probability Distribution
As always in order to calculate standard deviation, we must start by calculating the
variance. The population variance for a discrete probability distribution is:
 2   ( x   ) 2 P( x)   x 2 P( x)   2
The population standard deviation is the square root of the variance.
The notation follows:
  2

is a lowercase Greek letter said as ‘sigma’ (Can be any nonnegative real number)
Example 3
Roll a die and X is the number rolled. Find the standard deviation of X.
Example 4
This is a realistic example where at least the mean is an important value. Typically
standard deviation is not of much use in these situations. However, it is used as a unit of
measure in more complex statistical analysis. It is important you understand the idea so
that you will understand how standard deviation is used in more advanced analysis which
will be done in the coming material.
Suppose we sell life insurance policies. We will look at a $100,000 one year term policy.
The random variable of interest in this case is X = payout of the policy. The payout is
$100,000 if death occurs in the next year and $0 otherwise. When selling life insurance, a
company figures out the probability of payout based on life expectancy. This is typically
estimated based on a medical examination and/or a medical questionnaire. Suppose the
policy has the following probability distribution.
X
$0
$100,000
P(X)
0.995
0.005
Find the mean and standard deviation of X. What does the mean tell you in this example?