01
Conditional density from joint density
Suppose that
and have joint probability density given by: (a) Compute
, for . (b) Compute
.
02
From conditional to joint, and back again
Suppose we have the following data about random variables
and : (a) Find the joint distribution
. (b) Find
.
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.
Solution
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 .
Solution
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 .
Solution