Linear Transformations and Combinations

person buys, X, is a random variable. Suppose we want an expression for the amount of money
spent by that particular person. Letting Y denote the amount of money spent, we can find a
function relating Y to X. If a person buys X tickets, then they spent 2X dollars on tickets. In
addition, they spent $5 to enter the carnival. The total amount spent is Y = 2X + 5. Since Y has
the correct form (aX + b, with a = 2, and b = 5), we say Y is a linear transformation of X.
If we know the mean and variance of X, then there are simple formulas to compute the mean
and variance of a linear transformation Y . We derive the formulas for discrete random variables.
We know that E[X] =
x xP (X = x) and E[h(X)] = x h(x)P (X = x). A linear transformation
is a particular form of h(x), so
The formula applies because E[X] =
x
random variable.
E[aX + b] =
x (ax + b)P (X = x)
=
x axP (X = x) + bP (X = x)
=
x axP (X = x) + x bP (X = x)
= a
x xP (X = x) + b x P (X = x)
= aE[X] + b
xP (X = x) by definition and x P (X = x) = 1 for any
V [aX + b] = E[((aX + b) − (aE[X] + b)) 2 ]
= E[(aX + b − aE[X] − b) 2 ]
= E[a 2 (X − E[X]) 2 ]
= a 2 E[(X − E[X]) 2 ]
= a 2 V [X]
These formulas E[aX + b] = aE[X] + b and V [aX + b] = a 2 V [X] apply for any random variable,
discrete or continuous (we did not derive the calculus, but integrals have many of the properties of
summations, such as factoring out constants and breaking sums into two parts, so it makes sense).
The expectation formula is simple to remember, the expectation of a linear transformation is
the same linear transformation of the expectation. The variance requires a little more explanation.
Variance is a measure of spread. Adding a constant b doesn’t change the spread, so it can be
ignored in computed the variance. When we multiply by a, we have to remember the variance is
in squared units, so the constant a is squared in the formula.
Suppose for our carnival example any particular person buys an average of 5 tickets with a
variance of 9 tickets. What is the mean and variance of the amount of money spent by a particular
person? We said Y = 2X + 5, so E[Y ] = 2E[X] + 5 = 2(5) + 5 = 15 and V [Y ] = 2 2 (9) = 36.
3 Linear Combinations
Often we deal with several random variables at once. A linear combination of two random variables
X and Y has the form aX + bY + c, where a, b, and c are fixed constants found from the problem.
Linear combinations can involve many random variables. A linear combination Y of n random
variables X1, . . . , Xn is
Y = a1X1 + a2X2 + . . . + anXn + b
where the ai and b coefficients are fixed values which, again, are found in the problem.
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