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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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