Page 445 - Design for Six Sigma for Service (Six SIGMA Operational Methods)

Statistical Basics and Six Sigma Metrics 403

normal population will locate within a distance of one standard deviation
from the mean. Similarly, P(m − 2s ≤ y ≤ m +2s) = P(–2 ≤ z ≤ 2) = 0.9545 =
95.45%, that is; 95.45 percent of observations from a normal population
will locate within a distance of two standard deviations from the mean. In
addition, P(m − 3s ≤ y ≤ m +3s) = P(–3 ≤ z ≤ 3) = 0.9973 = 99.73%; that is,
99.73 percent of observations from a normal population will locate within
three standard deviations distance from the mean.

Exponential Distribution
An exponential distribution is featured by the following probability density
function:
− y/b
e
fy() = for 0 ≤ y < ∞
b
with a mean and variance of

E(y) = m = b and Var(y) = b 2

The exponential distribution is often used to model the following:
1. Lifetime of some electronic components
2. Interarrival time of customers entering a service facility
3. Time between consecutive machine failures or earthquakes

Binomial Distribution
The binomial distribution is a discrete probability distribution that charac-
terizes a binomial random variable. The binomial random variable can be
used for the following situation:
1. There are n successive trials. Each trial will only have two distinct
outcomes, S (success) or F (failure).
2. The probability of success P(S) = p, and P(F) = 1 – P(S) = 1 – p.
3. The result of each trial will not affect the results of any other trials.

If this situation is true, then the number of successes y out of n trials will be
a binomial random variable and its probability function p(y) will be

−
py() = n! p ( − p) n y y = 0, 1,…,n
y
1
−
yn y)!
!(
The mean and variance of the binomial random variable are
2
E(y) = m = np and Var(y) = s = np(1 – p)