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© Sven Thommesen 2011
Chapter 6: Continuous probability distributions
[Edited 10/05/08]
Recall from Chapter 5 that we used the
probability mass function
f(x
i
) to
assign a probability Pr(x=x
i
) to every possible outcome x
i
of a discrete
random variable X. We could do this because the number of possible
outcomes was limited.
A
discrete probability function
is a probability function which assigns
probabilities to possible values of a discrete variable “x”. That is, our
experiment produces outcomes that are whole numbers (integers), and the
pmf assigns a probability value to every possible integer outcome.
A discrete pmf obeys the general rules of probability this way:
the probability of any specific outcome must be between 0 and 1
the sum of probabilities
for all possible values of “x” (all elements in
the sample space) must be equal to 1.
the probability of any “x” that is not in the sample space is 0.
For the pmf, we write: Pr(x = x
0
) = f(x
0
)
We have discussed different examples of discrete pmf’s:
the discrete uniform probability function
the binomial probability function
the Poisson probability function
the hypergeometric probability function
… and there are many more, each suited for use in some particular set of
circumstances.
[For statisticians, part of the job is to determine which pmf to use for a
given problem.]
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PDF: the Probability Density Function
A
continuous probability function
is a probability function which assigns
probabilities to possible values of a continuous (real) variable “x”
.
When the variable x is a continuous variable (consisting of real numbers),
assigning probability to a specific value of x is no longer possible, since
between any two real numbers (say, 1.0 and 2.0) there is literally an infinite
set of additional real numbers! For the simplest case of a uniform probability
function, the probability of a given value for x would then have to be
P
(x) = 1/N = 1/∞ = 0!
So for continuous probability functions we need to interpret things a little
differently.
For continuous variables, we do not talk about the probability that x will be
equal to a specific value x
0
(always equal to zero), but rather about
the
probability that x will fall in the interval between two values
x
0
and
x
1
, written
Pr(x
0
≤ x ≤ x
1
)
We still draw a probability function, here called the
probability density
function
or PDF, corresponding to our relative frequency graphs
in chapters
3 and 5.
[Diagrams: a bell curve with limits x
0
and x
1;
the Exponential pdf]
In the diagrams above, we interpret the probability of the event “x falls
between x
0
and x
1
” as
the area under the probability density curve
between x
0
and x
1
.
Note that the
sample space
for a continuous probability function must be
an
interval
on the real line: it can be (
∞, +∞), or [0, +∞), or [a,b] for
arbitrary values a and b.
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 Spring '09
 Business, Normal Distribution, Probability, Probability theory, probability density function, CDF

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