01
Digit of a real number
Suppose a real number is chosen randomly in the unit interval
. Consider the decimal expansion of this number. Let be a random variable giving the first digit after the decimal point. Find the possible values, the PMF, and the CDF of .
Solution
10
(1) Find the possible values of
. After the decimal point, any of the 10 digits can appear, so
.
(2) Find the PMF of
. Each digit has a
chance of appearing. So, the PMF of is
(3) Find the CDF of
. The CDF of
is given as follows.
(4)
Listing out the individual cumulative probabilities for each
Link to originalis also acceptable.
02
Gambling with a coin
Two players, A and B, are flipping a fair coin together. If it comes up heads, A pays $1 to B, and if it comes up tails, B pays $1 to A.
They play five rounds. Let
be a random variable recording A’s final winnings. (a) Find the set of possible values of
. (I.e., the set of outcomes with nonzero probability.) (b) Find the PMF and CDF of
.
Solution
11
(a)
Describe the situation.
Let
be A’s (not total) winnings after the round. Since A either loses one dollar or gains one dollar each round,
.
, so .
(b)
(1) Describe the PMF of
.
if and only if all 5 rounds are tails, so .
if and only if all 5 rounds are heads, so .
if 4 rounds are tails, and if 4 rounds are heads. Since the coin is fair, .
if 3 rounds are heads, and if 3 rounds are tails. Since the coin is fair, .
(2) Define the PMF of
.
(3) Define the CDF of
. Note that the jumps in the PDF are the possible discrete values of
. Link to original
03
PMF from CDF for tracking gasoline
I am trying to keep track of how much gasoline (rounded to the nearest integer and denoted by
) my car uses every week. I have managed to find the CDF of : Assume gasoline costs $3 a gallon. Let
denote the amount of money I spend on gasoline every week. What is the PMF of ?
Solution
18
First find the PMF of
:
0 1 2 3 0.1 0.1 0.3 0.5 Now the PMF of
: Link to original
0 3 6 9 0.1 0.1 0.3 0.5
04
Patients in the hospital
After being discharged from the hospital following a particular surgery, patients often make visits to their local emergency room for treatment. The function below is the CDF of X, the number of emergency room visits per patient:
(a) Find the probability a patient will make more than 1 visit to the emergency room.
(b) Find the probability a patient will not visit the emergency room.
Solution
17
(a)
(b)
Link to original
05
Construct random variable from PMF
The probability space
is the set of infinite strings of or , for example . There is a probability measure determined by the interpretation of or as the outcome of a fair coin flip. For a subset containing all strings with specified (given) values of the first letters, then . E.g. if is everything starting with , then . (a) Construct a random variable
on such that the PMF of is given by: (b) Repeat (a) but with a biased coin for which
.
06
Variance from CDF: Drill bit changes
The bits for a particular kind of drill must be changed fairly often. Let
denote the number of holes that can be drilled with one bit. The CDF of is given below: (a) Find the probability that a bit will be able to drill more than 2 holes.
(b) Find
by constructing the PMF.
Solution
10
(a)
(b)
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07
Half the babies are female?
At Grace Community Hospital, 4 babies are delivered in one day. At Hope Valley Hospital, 6 babies are delivered in one day.
Consider these two events:
- (i) exactly half the babies born at Grace Community are female
- (ii) exactly half the babies born at Hope Valley are female
Perform a calculation to determine whether Event (i) is more probable, Event (ii) is more probable, or they are equally probable. (Assume the probability of each baby being born male or female is
.)
Solution
12
(i):
(ii):
So event (i) is more probable.
Link to original