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)
Solution
11
(a)
Note that the formula for
.
(b)
Compute
using the same formula.
(c)
(1) Recall the formula for
.
(2) Compute
and .
(3) Compute
.
(4) Compute
.
(d)
(1) Recall the formula for
.
(2) Compute
and .
(3) Compute
and .
(4) Compute
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02
Covariance etc. from independent densities
Suppose
and are independent variables with the following densities: Compute:
(a)
(b) (c) (d)
Solution
05
(a)
Compute
(b)
Compute
.
- Note that
and .
(c)
Compute
.
- Since
and are known to be independent, .
(d)
Compute
Link to original
- Since
, .
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?
Solution
09
(1) Define random variables to describe the problem.
- Let
to represent the arrival time of the plumber. - Let
represent the completion time of the sink fix.
(2) Compute
- This represents the expected time the plumber finishes the job.
- Thus, we expect the plumber to finish at
.
(3) Compute the variance of the finish time.
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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?
Solution
06
(1) Define random variables for partitioning the 30 flips into groups of 10.
Let
be the number of heads in the first 10 flips. Let
be the number of heads in the middle 10 flips. Let
be the number of heads in the last 10 flips. Clearly,
and are independent. Note that
and .
(2) Compute
and
(3) Compute
(4) Compute
Since
and , . Thus,
(5) Compute
(6) Compute
. Link to original
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
.
Solution
07
(1) Define indicator variables
and Let
denote whether occurs Let
denote whether occurs. Note that they are independent since
and are independent ( ). Let
.
(2) Compute
. Link to original
06
When Suppose
for two random variables and . Show that
, where . Find the formula for
. Hint: Study the derivation that
, and think about . (Note: A similar result and argument holds for
.)
Solution
08
(1) Recall the formula for
. Therefore, if
, then , and note that .
(2) Compute
. Thus,
.
(3) Isolate
in the above equation. Thus,
Link to original, where .
07
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.)
Solution
10
(a)
Compute
by integrating the joint PDF.
(b)
Compute
.
(c)
Compute
(d)
(1) Compute
(2) Compute
.
(e)
(1) Compute
(2) Compute
.
(f)
Compute
(g)
Compute
.
(h)
Determine independence.
Link to original
- Since
, we can conclude that and are not independent.