Page 510 - Discrete Mathematics and Its Applications

7.4 Expected Value and Variance 489

We conclude that

7
91 2 35
V(X) = − = . ▲
6 2 12
EXAMPLE 16 What is the variance of the random variable X((i, j)) = 2i, where i is the number appearing
on the first die and j is the number appearing on the second die, when two fair dice are rolled?

Solution: We will use Theorem 6 to find the variance of X. To do so, we need to find the
2
expected values of X and X . Note that because p(X = k) is 1/6 for k = 2, 4, 6, 8, 10, 12 and
is 0 otherwise,

E(X) = (2 + 4 + 6 + 8 + 10 + 12)/6 = 7,

and

2 2 2 2 2 2 2
E(X ) = (2 + 4 + 6 + 8 + 10 + 12 )/6 = 182/3.
It follows from Theorem 6 that

2
2
V(X) = E(X ) − E(X) = 182/3 − 49 = 35/3. ▲
Another useful property is that the variance of the sum of two or more independent random
variables is the sum of their variances. The formula that expresses this property is known as
Bienaymé’s formula, after Irenée-Jules Bienaymé, the French mathematician who discovered it
in 1853. Bienaymé’s formula is useful for computing the variance of the result of n independent
Bernoulli trials, for instance.

THEOREM 7 BIENAYMÉ’S FORMULA If X and Y are two independent random variables on
a sample space S, then V(X + Y) = V(X) + V(Y). Furthermore, if X i ,i = 1, 2,...,n,
with n a positive integer, are pairwise independent random variables on S, then
V(X 1 + X 2 + ··· + X n ) = V(X 1 ) + V(X 2 ) + ··· + V(X n ).

IRENÉE-JULES BIENAYMÉ (1796–1878) Bienaymé, born in Paris, moved with his family to Bruges in
1803 when his father became a government administrator. Bienaymé attended the Lycée impérial in Bruges,
and when his family returned to Paris in 1811, the Lycée Louis-le-Grand. As a teenager, he helped defend Paris
during the 1814 Napoleonic Wars; in 1815, he became a student at the École Polytechnique. In 1816 he joined
the Ministry of Finances to help support his family. In 1819, he left the civil service, taking a job lecturing
mathematics at the Académie militaire de Saint-Cyr. Unhappy with conditions there, he soon returned to the
Ministry of Finances. He attained the position of inspector general, remaining until forced to retire in 1848
for political reasons. He was able to return as inspector general in 1850, but he retired a second time in 1852.
In 1851 he briefly was professor at the Sorbonne and also served as an expert statistician for Napoleon III.
Bienaymé was one of the founders of the Société Mathématique de France, and in 1875 was its president.
Bienaymé was noted for his ingenuity, but his papers frustrated readers by omitting important proofs. He published sparsely,
often in obscure journals. However, he made important contributions to probability and statistics, and to their applications to the
social sciences and to finance. Among his important contributions are the Bienaymé-Chebyshev inequality, which provides a simple
proof of the law of large numbers, a generalization of Laplace’s least square method, and Bienaymé’s formula for the variance of a
sum of random variables. He studied the extinction of aristocratic families, declining despite general population growth. Bienaymé
was a skilled linguist; he translated the works of Chebyshev, a close friend, from Russian to French. It has been suggested that his
relative obscurity results from his modesty, his lack of interest in asserting the priority of his discoveries, and the fact that his work
was often ahead of its time. He and his brother married two sisters who were daughters of a family friend. Bienaymé and his wife
had two sons and three daughters.