Advanced High-School Mathematics

SECTION 6.1 Discrete Random Variables 321

Next, we define thevarianceσ^2 (or Var(X)) of the random variable
X having mean μ by settingσ^2 =E((X−μ)^2 ). The standard de-
viation σis the non-negative square root of the variance. The mean
and variance of a random variable are examples ofparametersof a
random variable.

We shall derive an alternate—and frequently useful—expression for
the variance of the random variableXwith meanμ. Namely, note that

Var(X) = E((X−μ)^2 )

= E(X^2 − 2 μX+μ^2 )

= E(X^2 )− 2 μE(X) +μ^2 (by (6.3))

= E(X^2 )−μ^2. (6.4)

We turn now to the variance of the discrete random variableX+Y.
In this case, however, we require thatXandY areindependent. This
means that for all valuesxandywe have

P(X=xandY =y) =P(X=x)P(Y =y).^4

In order to derive a useful formula for Var(X+Y), we need the result
that givenX andY are independent, thenE(XY) =E(X)E(Y); see
Exercise 1, below. Using (6.4), we have

Var(X+Y) = E((X+Y)^2 )−μ^2 X+Y

= E((X+Y)^2 )−(μX+μY)^2

= E(X^2 + 2XY +Y^2 )−(μX+μY)^2

= E(X^2 ) +E(2XY) +E(Y^2 )−(μ^2 X+ 2μXμY+μ^2 Y)

= E(X^2 )−μ^2 X+ 2E(X)E(Y)− 2 μXμY +E(Y^2 )−μ^2 Y

= Var(X) + Var(Y). (6.5)

(^4) An equivalent—and somewhat more intuitive—expression can be given in terms of conditional
probabilities. Namely, two eventsAandBare equivalent precisely whenP(A|B) =P(A). In terms
of discrete random variablesXandY, this translates intoP(X=x|Y =y) =P(X=x) for any
possible valuesxofXandyofY.