Description
problem 1. Suppose we roll a fair die three times, independently.
(a) Find the expectation of the number of sixes appearing in the three rolls.
(b) Find the expectation of the number of odd numbers appearing in the three rolls.
Solution:
problem 2. In homework 5, we worked with a continuous random variable X with the following PDF:
fX(x) = (
x
2
3
if x ∈ [−1, 2]
0 otherwise
(a) Compute Ex[X].
There is a powerful theorem nicknamed ”The Law of the Unconscious Statistician” (LOTUS) that states:
Ex[g(X)] = Z ∞
−∞
g(x)fX(x)dx
(b) Use LOTUS to compute Ex h
1
X
i
.
(c) Is Ex h
1
X
i
= 1
Ex[X]
?
(d) Use LOTUS to compute Ex[X
2 + 1].
Solution:
problem 3. A building has 10 floors above the basement and an elevator. Suppose 12 people get into the
elevator at the basement, and each chooses a floor uniformly at random to get out, independently of the
others. Find the expected number of floors where the elevator makes a stop to let out at least one person.
Solution:
problem 4. Suppose that the birthday of each of three people is equally likely to be any one of the 365
days of the year, independently of others. Let Bij denote the event that person i has the same birthday as
person j, where the labels i and j may be 1, 2 or 3.
(a) Are the events B12 and B23 independent?
(b) Are the events B12, B23, and B13 pairwise independent?
(c) Are the events B12, B23, and B13 mutually independent?
Solution:
problem 5. The Great Snyderinsky, a local magician, asks the audience to toss two coins, without showing
the result to him, and to write their results (either ‘H’ for heads or ‘T’ for tails) in two separate envelopes
labelled A and B respectively. He also asks the audience to take an extra envelope labelled C and write ‘H0
in it if the previous two coin tosses gave the same result, and write ‘T
0
in it if the previous two coin tosses
were different from each other (e.g. ‘H‘ ’T’). He then tells the audience to show him the content of any two
envelopes, and claims that he can use his magical moustache to guess the content of the third envelope.
Explain how the trick works in terms of independence. In particular address the following questions. Let
XA =
(
1 if envelope A contains HEADS
0 if envelope A contains TAILS
XB =
(
1 if envelope B contains HEADS
0 if envelope B contains TAILS
XC =
(
1 if envelope C contains HEADS
0 if envelope C contains TAILS
are XA, XB and XC mutually independent? Are they pairwise independent?
problem 6. Download the HW 8 Jupyter notebook from Piazza. Complete all the exercises in the notebook. Submit the Jupyter notebook with your solutions to the Homework 8 Programming assignment on
Gradescope. Your submission should be a single .ipynb file.
H8-2
