01
Conditional distribution and expectation from joint PMF
Suppose that
and have the following joint PMF: Notice that the possibilities for
depend on the choice of . First, show that
. Then compute: (a)
(b) (c) (d)
Solution
06
Find possible combinations of
and .
- For a given value of
, there are possible options for . - Thus, in total, there are
possible combinations. - Since
, we have that . (a)
We have that
Thus,
(b)
(1) State formula
(2) State final answer
(c)
Compute expectation:
(d)
Find formula for expectation.
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02
Conditional distribution and expectation from joint PDF
Suppose that
and have the following joint PDF: Notice that the range of possibilities for
depends on the value of . First, show that
. Then compute: (a)
(b) (c) (d)
Solution
07
Integrate joint PDF and solve for
. (a)
Integrate joint PDF with respect to
to obtain
(b)
Use formula for conditional distribution.
(c)
(1) Plug in
into the formula found in part (b)
(2) Integrate to find expectation.
(d)
Integrate formula in part (b) with respect to
. Link to original
03
“Plug In” Expectation Identity
Suppose
is a function, and and are two random variables. Verify this formula in the continuous case, using the definitions:
Using that formula, prove this formula:
for two functions
and and random variables and . Notice that here the expectations are viewed as random variables. Hint for second question: Both sides are functions of
. Write these functions as and and check equality of the functions.
Solution
08
(1) State formula for expectation given
.
(2) Note that
is fixed inside the conditional expectation.
- Treat
as inside integral.
(3) Prove the second formula.
(4) State final answer.
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04
Expectation Multiplication Rule
Prove the following identity using Iterated Expectation along with the previous exercise:
Note: The solution is short once you find it – please clearly identify your choices for
and functions.
Solution
09
Let
and . Link to original
05
How many customers buy a cake?
Let
count the number of customers that visit a bakery on a random day, and assume . Let
count the number of customers that make a purchase. Each customer entering the bakery smells the cakes, and this produces a probability of buying a cake for that customer. The customers are independent. Find
. Are and positively or negatively correlated? Hint: Compute
, and use this to compute in terms of . Now deduce using Iterated Expectation. Finally, compute using the Expectation Multiplication Rule from the previous exercise. Now put everything together to find .
Solution
10
(1) Find
. Note that this follows a binomial distribution with parameters
. Thus,
(2) Find
using iterated expectation.
(3) Find
(4) Find covariance
(5) State final conclusions. Since the covariance is positive,
Link to originaland are positively correlated.