Page 233 - DMGT524_TOTAL_QUALITY_MANAGEMENT

Total Quality Management

Notes It states that the probability of r non-conforming items in a sample of size n is equal to the
product of the possible combinations of nonconforming items, times the possible combinations
of conforming items, divided by the possible combinations of samples of n that can be drawn
from lots of size N.
It we use the hypergeometric probability law to calculate the probabilities of 0, 1, 2 or 3 defective
article as was done in the previous section, we would get the same values.

Probability Density Function

The probability density function of a continuous random variable is a function which can be
integrated to obtain the probability that the random variable takes a value in a given interval.

More formally, the probability density function, f(x), of a continuous random variable X is the
derivative of the cumulative distribution function F(x):

d
f(x) = F(x)
dx
Since F(x) = P(X ≤ x) it follows that:
Ú f(x)dx = F(b) – F(a) = P(a < X < b)

If f(x) is a probability density function then it must obey two conditions:
(a) that the total probability for all possible values of the continuous random variable X is 1:

Ú f(x)dx = 1

(b) that the probability density function can never be negative: f(x) > 0 for all x.

14.6.6 Binomial Distribution

In case of hypergeometric probability distribution the calculations of probability took into
account the change in probabilities with every draw. However, it can be assumed that if the lot
size is infinite, the probabilities remain constant over draws. Such problems can be solved using
the binomial probability distribution.
Typically, a binomial random variable is the number of successes in a series of trials, for example,
the number of ‘heads’ occurring when a coin is tossed 50 times.
A discrete random variable X is said to follow a Binomial distribution with parameters n and p,
written X ~ Bi(n,p) or X ~ B(n,p), if it has probability distribution:

n
⎛⎞
−
x
P(X = x) = ⎜⎟ p (1 p)
⎝⎠
x
where x = 0, 1, 2, ..., n
n = 1, 2, 3, ...
p = success probability; 0 < p < 1
⎛⎞ n!
n
⎜⎟ = x!(n − x)!
⎝⎠
x

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