Functions of two variables, covariance, correlation
01
Covariance and correlation
The joint PMF of
and is given by the table:
0 1 2 3 1 2 3 0 Compute:
(a)
(b) (c) (d)
02
Covariance etc. from independent densities
Suppose
and are independent variables with the following densities:
Compute:
(a)
(b) (c) (d)
03
Plumber completion time
A plumber is coming to fix the sink. He will arrive between 2:00 and 4:00 with uniform distribution in that range.
Sink fixes take an average of 45 minutes with completion times following an exponential distribution.
When do you expect the plumber to finish the job?
What is the variance for the finish time?
04
Correlation between overlapping coin flip sequences
Suppose a coin is flipped 30 times.
Let
count the number of heads among the first 20 flips, and count the heads in the last 20. Find
. Hint: Partition the flips into three groups of 10. Create three variables, counting heads, and express
and using these. What is the variance of a binomial distribution?
05
Variance puzzle: indicators
Suppose
and are events satisfying: Let
count the number of these events that occurs. (So the possible values are .) Find
. Hint: Try setting
.
06
When Suppose
for two random variables and . Show that
, where . Find the formula for
. Hint: Study the proof that
, and think about . (Note: A similar result and argument holds for
.)
07
Further practice: Covariance etc. from joint density
Suppose
and are random variables with the following joint density:
Compute:
(a)
(b) (c) (d) (e)
(f) (g) (h) Are and independent? (It is worth thinking through which of these can be computed in multiple ways.)