Inference based on discrete observations
.
Example
.
.
The treasury of an underdeveloped country produces
coins whose probability of heads is a r.v.
P
with PDF
f
P
(
p
) =
pe
p
if
p
∈
[
0
,
1
]
0
,
o.w.
.
(a)
Find the probability that a coin toss results in heads.
(b)
Given that a coin toss resulted in heads, find the
conditional PDF of
P
. (Exercise:
What if the coin is
tossed twice and heads is observed once?)
(c)
Given that the first coin toss resulted in heads, find the
conditional probability of heads on the next toss.
74/75

Continuous Random Variables and PDFs
Cumulative Distribution Functions
Normal Random Variables
Multiple Continuous Random Variables
Conditioning
The Continuous Bayes’ Rule
Inference about a cont. r.v. based on cont. observations
Inference about a discrete r.v.
Inference based on discrete observations
.
Some Additional Problems
.
Example (The sum of a random number of r.v.s)
.
.
You visit a random number
N
of stores and in the
i
th
store,
spend a random amount of money
X
i
. The total amount of
money you spend is
T
=
X
1
+
X
2
+
. . .
+
X
N
where
N
is a positive integer r.v. with mean
E
[
N
]
and
variance
var
(
N
)
. The r.v.s
X
i
are independent and
identically distributed (i.i.d.), and have mean
E
[
X
]
and
variance
var
(
X
)
. Evaluate the mean and variance of
T
.
75/75

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- Fall '19