MATH1715 Tutee Revision Questions

Question: 29/12/2010
I am not quite sure how to obtain the answer for the following question.
The demand for a certain weekly magazine at a news stand is a random variable with
probability mass function pX(x) given by
x 3 4 5 6
pX(x) 0.20 0.44 0.32 0.04
The magazine sells for £1.50, and the cost to the owner is 50 pence per copy. Unsold copies
cannot be returned to the publisher.
(i) What is the expected weekly demand for this magazine?
(ii) If the owner orders three copies of the magazine, his net profit will be £3.00. What will
be his expected profit if he orders four copies?
(iii) How many copies should be ordered per week in order to maximise the expected profit?
Response: Here X denotes the number of copies demanded per week.
(i) E[X] = �
xpX(x) = µ say.
x
(ii) Let N denote the number of copies he orders. To see how to tackle the question, look
at some specific cases.
If N = 3, then he pays 3 × 50p = £1.50. He sells them all if X ≥ 3, which occurs with
probability unity. He receives 3 × 1.50 = £4.50, so his net profit is £3 as specified.
If N = 4, then he pays 4 ×50p = £2.00. There are two cases to now consider. If X = 3,
with probability 0.2, he sells 3, making net profit P3. If X ≥ 4, with probability 0.8, he sells
all 4, making net profit P4. His net profit P then satisfies E[P] = (0.2 × P3) + (0.8 × P4).
(iii) Repeat the calculations for N = 5 and N = 6. For example, for N = 5 you can
determine what he has spent buying the copies. For each of the cases X = 3, X = 4, X ≥ 5
you can determine his net profit and the associated probability. Thus you can determine
the mean profit if N = 5. Similarly for N = 6.
You could also consider the cases N = 0, 1, 2 and N = 7, 8, 9, . . .. In the latter case it
is easy to see that his costs increase while his return is capped by the maximum demand
X = 6; thus his net profit must certainly be monotonic decreasing for all N > 6. For N = 0
his net profit is zero (a safe risk-free place to be!). For 0 ≤ N ≤ 3 it can be seen that his
profit must be monotonic increasing as he always sells at least three copies. The maximum
must therefore lie between N = 3 and N = 6.
Question: 26/12/2010
Is it a general rule that Var[aX + bY ] = a Var[X] + b 2 Var[Y ] + 2ab cov(X, Y )?
Response: Yes.
Question: 26/12/2010
We toss a coin repeatedly until we see each side of the coin at least once. What is the mean
number of tosses required?
Response: Let X denote the number of tosses until we see each side at least once. What
is the distribution of X? What is E[X]?
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