Class summary:
Monday, 4/14
I began Chapter 4 (Continuous distributions) with an introduction to
the concept of a continuous random variable (as opposed to discrete
r.v.'s), and
two functions associated with a continuous r.v.,
the (probability) density function f(x)
and the cumulative distribution
function (c.d.f.) F(x). Most formulas that hold for discrete r.v.'s
have a continuous analog obtained by replacing sums by integrals and the
probabilities P(X=x) by the density function f(x). However, there are
some differences, the most important one being that
the probabilities P(X=x) which determined the distribution in the discrete
case, are all 0 for continuous r.v.'s X, and therefore cannot be used to
describe the distribution of X. The density function f(x) serves
as a substitute for P(X=x), but it is defined differently, namely as
the derivative of the c.d.f. F(x).
I introduced three important named continuous distribution, the uniform
distribution on an interval (a,b), the exponential distribution,
and the normal distribution.
I began a series of examples illustrating these concepts.
The problems
can be classified into two distinct types:
- (I) Given a description of a contiuous r.v., find
its density f(x). This (usually)
requires computing first F(x) (via the
defiition F(x)=P(X<x)), and taking the derivative to get f(x).
- (II) Given the density function f(x) of a r.v.,
compute probabilities involving this r.v., expectations, etc.
In the case of continuous r.v.'s this (usually) boils down
to the computation of integrals.
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