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134
Part III: Distributions and the Central Limit Theorem
Probability Distribution for X = Number of Dogs
Table 8-1
Owned by Apartment Renters

p(x)
x
0.50
0
1
0.40
0.07
2
0.03
3
The mean and variance of a
discrete random variable
The mean of a random variable is the average of all the outcomes you would
expect in the long term (over all possible samples). For example, if you roll a
die a billion times and record the outcomes, the average of those outcomes is
3.5. (Each outcome happens with equal chance, so you average the numbers
1 through 6 to get 3.5.) However, if the die is loaded and you roll a 1 more
often than anything else, the average outcome from a billion rolls is closer
to 1 than to 3.5.
The notation for the mean of a random variable X is (pronounced “mu
sub x”; or just “mu x”). Because you are looking at all the outcomes in the long
term, it’s the same as looking at the mean of an entire population of values,
which is why you denote it and not . (The latter represents the mean of
a sample of values [see Chapter 5].) You put the X in the subscript to remind
you that the variable this mean belongs to is the X variable (as opposed to a Y
variable or some other letter).
The variance of a random variable is roughly interpreted as the average
squared distance from the mean for all the outcomes you would get in the
long term, over all possible samples. This is the same as the variance of the
population of all possible values. The notation for variance of a random vari-
able X is . You say “sigma sub x, squared” or just “sigma squared.”
The standard deviation of a random variable X is the square root of the vari-
ance, denoted by (say “sigma x” or just “sigma”). It roughly repre-
sents the average distance from the mean.
Just like for the mean, you use the Greek notation to denote the variance and
2
standard deviation of a random variable. The English notation s and s repre-
sent the variance and standard deviation of a sample of individuals, not the
entire population (see Chapter 5).

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