Page 442 - Design for Six Sigma for Service (Six SIGMA Operational Methods)

400 Chapter Eleven

random variable if it can take only a countable number of values. A simple
example of a discrete random variable is the number of points that a fair six-
sided die will show on a toss, y. It can only be 1, 2, 3, 4, 5, or 6, with equal prob-
abilities. A random variable can also be a continuous random variable if it can
take all real numbers in a given interval. For example, a random person’s height
(that is, a person you randomly met on a street) is a continuous random variable.
The probability structure of a random variable, say y, is described by its
probability distribution. If y is a discrete random variable, then its prob-
ability distribution is described by the probability function, often denoted
by p(y). If y is a continuous random variable, its probability distribution is
described by its probability density function, often denoted by f(y).

The properties of the probability function p(y) and probability density
function f(y) are summarized as follows:
Discrete random variable y

0 ≤ p (y ) ≤ 1 y ’s are the possible values that y can take
i
i
P(y = y ) = P(y )
i
i
∑ py() = 1
i
All − y i
Continuous random variable y
fy() ≥ 0
Pa y b) = ∫ a b f () y dy
(
+∞
∫ −∞ fy
()dy = 1

11.3.1 Expected Value, Variance, and Standard Deviation

The random variable and probability distribution deal with the population.
The mean of the population (or population mean m) is the most frequently
used measure of central tendency for the population; m is also called the
expected value of the random variable y.

The expected value E(y) = m is defined as follows:

⎧ ∑ yp y( ) if y is discrete
⎪All
m = Ey() = ⎨ −y
⎪ ∫ +∞ yf y dy if y is continuous
()
⎩ −∞