Page 231 - DMGT524_TOTAL_QUALITY_MANAGEMENT

Total Quality Management

Notes Continuous Random Variables

A continuous random variable is one which takes an infinite number of possible values.
Continuous random variables are usually measurements. Examples include height, weight, the
amount of sugar in an orange, the time required to run a mile.
A continuous random variable is not defined at specific values. Instead, it is defined over an
interval of values, and is represented by the area under a curve (in advanced mathematics, this is
known as an integral). The probability of observing any single value is equal to 0, since the
number of values which may be assumed by the random variable is infinite.

Suppose a random variable X may take all values over an interval of real numbers. Then the
probability that X is in the set of outcomes A, P(A), is defined to be the area above A and under
a curve. The curve, which represents a function p(x), must satisfy the following:
1. The curve has no negative values, (p(x) > 0 for all x)
2. The total area under the curve is equal to 1.
A curve meeting these requirements is known as a density curve.

14.6.5 Probability Distribution

A probability distribution describes the values and probabilities associated with a random event.
The values must cover all of the possible outcomes of the event, while the total probabilities
must sum to exactly 1, or 100%. For example, a single coin flip can take values Heads or Tails with
a probability of exactly 1/2 for each; these two values and two probabilities make up the
probability distribution of the single coin flipping event. This distribution is called a discrete
distribution because there are a countable number of discrete outcomes with positive probabilities.
A continuous distribution describes events over a continuous range, where the probability of a
specific outcome is zero. For example, a dart thrown at a dartboard has essentially zero probability
of landing at a specific point, since a point is vanishingly small, but it has some probability of
landing within a given area. The probability of landing within the small area of the bulls eye
would (hopefully) be greater than landing on an equivalent area elsewhere on the board.
A smooth function that describes the probability of landing anywhere on the dartboard is the
probability distribution of the dart throwing event. The integral of the probability density function
(pdf) over the entire area of the dartboard (and, perhaps, the wall surrounding it) must be equal
to 1, since each dart must land somewhere.
The concept of the probability distribution and the random variables which they describe underlies
the mathematical discipline of probability theory, and the science of statistics. There is spread or
variability in almost any value that can be measured in a population (e.g. height of people,
durability of a metal, etc.); almost all measurements are made with some intrinsic error; in
physics many processes are described probabilistically, from the kinetic properties of gases to the
quantum mechanical description of fundamental particles. For these and many other reasons, simple
numbers are often inadequate for describing a quantity, while probability distributions are
often more appropriate models. There are, however, considerable mathematical complications
in manipulating probability distributions, since most standard arithmetic and algebraic
manipulations cannot be applied.

Discrete Probability Distribution

A probability distribution is called discrete if its cumulative distribution function only increases
in jumps.

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