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MSBA7003 Quantitative Analysis Methods
Assignment 1 (Due October 5 at 09:00 a.m.; Please submit your solutions with the template.)
Q1.
The joint probability distribution among three random variables (X, Y, and Z) is given in the
following tables.
Z = 0 Z = 1
X = 0 X = 1 X = 0 X = 1
Y = 0 0.18 0.12 Y = 0 0.064 0.096
Y = 1 0.18 0.12 Y = 1 0.096 0.144
Which of the following statement(s) is(are) true?
(A) X = 0 and Y = 0 are unconditionally independent.
(B) Given Z = 0, the two events X = 1 and Y = 1 are independent.
(C) E[ X – Y | Z = 1 ] = 0.096.
(D) P( Z = 1 | X = 0 ) = 0.16.
(E) None of the above.
Q2.
According to the Venn diagram below, which of the following statement(s) is(are) true?
(A) Event A and event B are dependent without knowing event C.
(B) Given that C occurred, A and B are independent.
(C) P(BC) = P(BC|A).
(D) Without knowing B, event A and event C are independent.
(E) None of the above.
Q3.
ABC Inc. is considering launching a new product and there are two options: product X and
product Y. Product X requires an initial investment of \$15 million and product Y requires \$5
million. The total profit (before subtracting the initial investment) that can be generated by
each product depends on the market condition. If the market is strong, product X can generate
a total profit of \$100 million and product Y can generate \$20 million; if the market is weak,
product X will lead to a net loss of \$80 million and product Y will cause a net loss of \$9 million
(before subtracting the initial investment). However, the company has no idea about the
probability of a strong market. The probability of a strong market can be 0.2 or 0.5 or 0.8, and
they are equally likely.
To get a better understanding of the market before deciding the choice of the new product, the
manager hired a consulting firm to conduct a market research. According to historical data, this
consulting firm can successfully predict a strong market in 65% cases and can correctly predict a
weak market in 55% cases. This time, their report gave a favorable conclusion.
Based on this information, which of the following statement(s) is(are) true?
(A) If the consulting firm is not hired, it is better not to introduce any product.
(B) Given the favorable report, the expected probability of a strong market is about 0.522.
(C) Given the favorable report, ABC should introduce product X.
(D) The expect value of the consulting firm’s report is about \$0.1259 million.
(E) None of the above.
Q4.
For the Google’s interview question, which of the following statement(s) is(are) true?
(A) The expected minimum number of tries is 3.
(B) The expected minimum number of tries is 4.
(C) The expected minimum number of tries is 5.
(D) The expected minimum number of tries is 6.
(E) None of the above.
Q5.
You are testing the quality of light bulbs made by a manufacturer. The life span follows an
exponential distribution with an unknown mean. There are three possibilities for the mean life
span: high quality (2 years), medium quality (1 year), and low quality (half a year). They are
equally likely at the beginning. You sampled three light bulbs for a half-year period. The first
one lasts for half a year and is still working; the second one lasted for less than one month (i.e.,
<1/12 years); the third one lasts for half a year and is still working.
Based on this information, which of the following statement(s) is(are) true?
(A) The mean life span is less than 1 year.
(B) The light bulbs are most likely to be of medium quality.
(C) At the beginning, the probability that a light bulb can last for more than half a year is 0.42.
(D) After observing the samples, the probability that a light bulb can last for more than half a
year is 0.45.
(E) None of the above.