Class summary:
Friday, 4/18
As a final example on continuous distributions,
I proved that the exponential distribution
satisfies P(T>s+t|P(T>s) = P(T>t) for all positive s,t.
This property is called the "no memory" property since it can be
interpreted as saying that an expoentially distributed r.v. has no
memory of its past, or shows no signs of aging.
I began the discussion of the so-called
Poisson process, which represents an interesting and important example
of a "stochastic process" involving both continuous and discrete random
variables. The Poisson process can be thought of as a sequence of
points placed randomly (according to some rules) on a time axis.
It serves as a model for real-world processes such as the burning out and
replacing of light bulbs; the arrival of customers at a store; or the
arrival of buses at a bus stop.
Associated with such a process are two types of random variables, one
discrete, the other continuous.
- The time intervals ("waiting times") W1, W2, W3, etc.,
between consecutive points.
- The number N(I) of points in a given interval I on the
time axis.
The Poisson process is characterized by the following properties:
- (I) The r.v.'s W1, W2, ..., are independent and have
exponential(lambda) distribution.
- (II) The r.v.'s N(I) have Poisson distribution with parameter
lambda t, where t is the length of I. Moreover, the N(I)'s
corresponding to disjoint intervals are independent.
These properties turn out to be equivalent, i.e., a process that
satisfies (I) automatically satisfies (II), and vice versa, and they
define the Poisson process uniquely.
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