Due date: Thursday 2/26, 11:59pm
Joint distributions
01
06
Link to originalStop lights
Let
be the number of stop lights and the number of red lights at which you must wait on your drive to grounds each day. The joint PMF of and is given below.
0 1 2 0 0.05 0.07 0.08 1 0.08 0.10 0.12 2 0.12 0.33 0.05 Find the marginal PMF of
and compute .
Solution
02
03
Link to originalMarginals from joint PMF
Suppose the discrete joint PMF of
and is given by:
Compute the marginal PMFs
and .
Solution
03
07
Link to originalGrad student mentors
A university lab has an incoming group of researchers who need mentors. There are 4 graduate students, 2 undergraduate students, and the lab director who could serve as mentors. Suppose that exactly 3 of these 7 people will be selected as mentors.
(a) How many different groups of 3 people could be chosen to be the three mentors?
(b) Suppose exactly 2 graduate students and 1 undergraduate student are selected to be the three mentors. How many different groups of 3 people could be selected?
Let
be the total number of graduate students chosen to be mentors, and be the total number of undergraduate students chosen to be mentors. (c) Construct the joint PMF of
and :
Solution
Independent random variables
04
04
Link to originalAlice and Bob meeting at a cafe
Alice and Bob plan to meet at a cafe to do homework together. Alice arrives at the cafe at a random time (uniform) between 12:00pm and 1:00pm; Bob independently arrives at a random time (uniform) between 12:00pm and 2:00pm on the same day. Let
be the time, past 12:00pm, that Alice arrives (in hours) and be the time, past 12:00pm, that Bob arrives (in hours). So and represent 12:00pm. Find the joint PDF of
and . (Hint: and are independent.)
Solution