Page 234 - DMGT524_TOTAL_QUALITY_MANAGEMENT

Unit 14: Statistical Process Control

The trials must meet the following requirements: Notes
(a) the total number of trials is fixed in advance;
(b) there are just two outcomes of each trial; success and failure;
(c) the outcomes of all the trials are statistically independent;

(d) all the trials have the same probability of success.
2
The Binomial distribution has expected value E(X), μ = np and Variance, σ = npq = np(1-p).
14.6.7 Poisson Distribution

Poisson distributions model some discrete random variables. Typically, a Poisson random variable
is a count of the number of events that occur in a certain time interval or spatial area. For
example, the number of cars passing a fixed point in a 5 minute interval, or the number of calls
received by a switchboard during a given period of time.
A discrete random variable X is said to follow a Poisson distribution with parameter m, written
X ~ Po(m), if it has probability distribution

m x
P(X = x) = e − m
x!
where x = 0, 1, 2, ..., n
m> 0.
The following requirements must be met:
(a) the length of the observation period is fixed in advance;

(b) the events occur at a constant average rate;
(c) the number of events occurring in disjoint intervals are statistically independent.
The Poisson distribution has expected value E(X) = m and variance V(X) = m; i.e. E(X) = V(X) = m.

The Poisson distribution can sometimes be used to approximate the Binomial distribution with
parameters n and p. When the number of observations n is large, and the success probability p
is small, the Bi(n, p) distribution approaches the Poisson distribution with the parameter given
by m = np. This is useful since the computations involved in calculating binomial probabilities
are greatly reduced.
14.6.8 The Normal Curve

Many phenomena in life where quantitative data is generated follow a normal curve. A normal
curve is shown in figure 14.17. This is a continuous density curve. The area under the curve is not
easy to calculate for a normal random variable X with mean μ and standard deviation σ. However,
tables (and computer functions) are available for the standard random variable Z, which is
computed from X by subtracting σ and dividing by σ.
A normal distribution has a bell-shaped density curve described by its mean μ and standard
deviation σ. The density curve is symmetrical, centered about its mean, with its spread determined
by its standard deviation. The height of a normal density curve at a given point x is given by

1 ⎛ x −μ ⎞ 2
1 − 2 ⎝ σ ⎟ ⎠
⎜
e
σ 2π

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