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Lecture 26
Agenda
1. Mixed random variables and the the importance of distribution function
2. Joint Probability Distribution for discrete random variables
Mixed random variables and the the impor
tance of distribution function
Any numerical quantity associated with a random experiment is called a
random variable. Mathematically we say, any function
X
:
S
→
R
is a
random variable. We have already deﬁned random variables like this in the
beginning of the course, and then we considered a special class of random
variables, namely the discrete random variables.
We know that for a discrete random variable
X
, when somebody asks me
about it’s probability distribution, I have to provide him two things
1. Range(X) =
{
x
1
,x
2
,x
3
,...
}
= The set of values that
X
takes.
2. and, for each
x
i
∈
Range
(
X
), I have to provide
P
(
X
=
x
i
).
Then we learned how to calculated the expectation, variance etc for a
discrete random variable and studied various examples like Binomial, Geo
metric, Negative Binomial .
.....
After that we considered continuous random variables, for which
P
(
X
=
x
) = 0 for all
x
∈
R
. We saw that if some one asked me about the distribution
of a continuous random variable
X
, I have to give him the density
f
X
:
R
→
[0
,
∞
), such that
1. for all
a < b
,
P
(
a < X < b
) =
R
b
a
f
X
(
x
)
dx
2. and,
R
∞
∞
f
X
(
x
)
dx
= 1.
Then similar to discrete random variables, we learned how to calculate
mean, variance etc for a continuous random variable. We also saw various
examples of continuous random variables like exponential, gamma, beta etc.
Now a natural question to ask would be —–
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 Fall '08
 Staff
 Probability

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