Due date: Friday 10/31, 9:00am
Conditional distribution
01
01
Link to originalConditional density from joint density
Suppose that
and have joint probability density given by: (a) Compute
, for . (b) Compute
.
02
02
Link to originalFrom conditional to joint, and back again
Suppose we have the following data about random variables
and : (a) Find the joint distribution
. (b) Find
.
03
03
Time till recharge
Let
denote the amount of time (in hours) that a battery on a solar calculator will operate properly before needing to be recharged by exposure to light. The function below is the PDF of . Suppose that a calculator has already been in use for 5 hours. Find the probability it will operate properly for at least another 2 hours.
Link to originalSolution
04
04
Cashews in a can
A nut company markets cans of mixed nuts containing almonds, cashews, and peanuts. Suppose the net weight of each can is exactly
, but the weight contribution of each type of nut is random. Let be the weight of almonds in a selected can and the weight of cashews. The joint PDF of and is given below: Suppose that the weight of cashews in a particular can is
. Calculate the probability that the weight of almonds in this can is more than . Link to originalSolution
05
05
New and returning customers
A sales representative will randomly select and call 2 customers. The representative’s goal is to get each customer to complete a satisfaction survey. Each of these customers is categorized as “new” or “returning.” 70% of customers are new and 30% are returning. Let
be the number of new customers that are called. (a) Construct the PMF of
, . The probability of any new customer completing the survey is 0.15, and the probability of any returning customer completing the survey is 0.20. (Customers operate independently.) Let
be the number of new customers that complete the survey. (b) Construct
, , and . (You should construct 3 separate PMFs.) (c) Construct the joint PMF of
and . Link to originalSolution
Conditional expectation
06
01
Link to originalConditional 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)
07
02
Link to originalConditional 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)
Optional challenge problems
You may do all of the next three problems instead of all of the problems above.
08
03
Link to original“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.
09
04
Link to originalExpectation 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.
10
05
Link to originalHow 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 .