Page 28 - Computational Statistics Handbook with MATLAB

14 Computational Statistics Handbook with MATLAB

Some events have a special property when they are considered together.
Two events that cannot occur simultaneously or jointly are called mutually
exclusive events. This means that the intersection of the two events is the
empty set and the probability of the events occurring together is zero. For
example, a piston ring cannot be both defective and good at the same time.
So, the event of getting a defective part and the event of getting a good part
are mutually exclusive events. The definition of mutually exclusive events
can be extended to any number of events by considering all pairs of events.
Every pair of events must be mutually exclusive for all of them to be mutu-
ally exclusive.

b
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a
y
Proob
Pr PPrr obob abbi ilit litlit lity yy
Probability is a measure of the likelihood that some event will occur. It is also
a way to quantify or to gauge the likelihood that an observed measurement
or random variable will take on values within some set or range of values.
Probabilities always range between 0 and 1. A probability distribution of a
random variable describes the probabilities associated with each possible
value for the random variable.
We first briefly describe two somewhat classical methods for assigning
probabilities: the equal likelihood model and the relative frequency method.
When we have an experiment where each of n outcomes is equally likely,
then we assign a probability mass of 1 n⁄ to each outcome. This is the equal
likelihood model. Some experiments where this model can be used are flip-
ping a fair coin, tossing an unloaded die or randomly selecting a card from a
deck of cards.
When the equal likelihood assumption is not valid, then the relative fre-
quency method can be used. With this technique, we conduct the experiment
n times and record the outcome. The probability of event E is assigned by
⁄
PE() = fn , where f denotes the number of experimental outcomes that sat-
isfy event E.
Another way to find the desired probability that an event occurs is to use a
probability density function when we have continuous random variables or
a probability mass function in the case of discrete random variables. Section
2.5 contains several examples of probability density (mass) functions. In this
text, f x() is used to represent the probability mass or density function for
either discrete or continuous random variables, respectively. We now discuss
how to find probabilities using these functions, first for the continuous case
and then for discrete random variables.
To find the probability that a continuous random variable falls in a partic-
ular interval of real numbers, we have to calculate the appropriate area under
the curve of f x() . Thus, we have to evaluate the integral of f x() over the inter-
val of random variables corresponding to the event of interest. This is repre-
sented by

© 2002 by Chapman & Hall/CRC