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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.

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