5. Functions of a Random Variable
Let X be a r.v defined on the model (, F , P ), and suppose
g(x) is a function of the variable x. Define
Y g ( X ).
Is Y necessarily a r.v? If so what is its PDF FY ( y ), pdf fY ( y ) ?
Clearly if Y is a r.v, then
8. One Function of Two Random
Given two random variables X and Y and a function g(x,y),
we form a new random variable Z as
Z g ( X , Y ).
Given the joint p.d.f f XY ( x , y ), how does one obtain f Z ( z ),
the p.d.f of Z ? Problems of thi
9. Two Functions of Two Random
In the spirit of the previous section, let us look at an
immediate generalization: Suppose X and Y are two random
variables with joint p.d.f f XY ( x, y). Given two functions g ( x, y )
and h( x, y ), define the ne
4. Binomial Random Variable Approximations
Conditional Probability Density Functions
Let X represent a Binomial r.v as in (3-42). Then from (2-30)
n k n k
P k1 X k 2 Pn ( k ) p q .
k k1 k
Since the binomial coefficient nk ( n nk! )!
10. Joint Moments and Joint Characteristic
Following section 6, in this section we shall introduce
various parameters to compactly represent the information
contained in the joint p.d.f of two r.vs. Given two r.vs X and
Y and a function g ( x, y
2. Independence and Bernoulli Trials
Independence: Events A and B are independent if
P ( AB ) P ( A) P ( B ).
It is easy to show that A, B independent implies A, B ;
A, B ; A, B are all independent pairs. For example,
B ( A A ) B AB AB and AB AB ,
3. Random Variables
Let (, F, P) be a probability model for an experiment,
and X a function that maps every , to a unique
point x R, the set of real numbers. Since the outcome
is not certain, so is the value X ( ) x . Thus if B is some
subset of R, we ma
6. Mean, Variance, Moments and
For a r.v X, its p.d.f f X (x) represents complete information
about it, and for any Borel set B on the x-axis
P X ( ) B
f X ( x ) dx .
Note that f X (x) represents very detailed information
7. Two Random Variables
In many experiments, the observations are expressible not
as a single quantity, but as a family of quantities. For
example to record the height and weight of each person in
a community or the number of people and the total income