6-11. LINEAR TRANSFORMATIONS OF RANDOM VARIABLES
Our goal is to determine the effect of various linear transformations (adding the same number to
each value or multiplying each number by some constant) on the various statistics such as mean,
variance, and standard deviation.
For each of the transformations described, put the "new" probability distribution in our expected
value calculator.
a. What effect does doubling each of the outcomes have on the expected
value, the variance, and the standard deviation?
E(X) =
o'x =
Ox=
b. What effect does adding 5 to each of the outcomes have on the
expected value, the variance, and the standard deviation?
E(X') =
o'x. =
Ox =
=
c. Now we are going to combine the effects of (a) and (b). First double X
and then add five. Before making any calculations, predict how the mean,
variance, and standard deviation will change. Then make your
calculations!
E(X') =
o²x₁ =
X 2 4
P(X) 0.3 02
4
P(X) 03
X"
P(X") 0.3
5
05
8 10
0.2 0.5
X 2 4 5
PX) 0.3 0.2 05
1
7 9 10
0.2 05
X 2 4
P(X) 03 0.2
9
P(X) 03
13
02
5
0.5
15
0.5