Assignment 1
ISyE 6644
1. A random variable Y has the following pdf:
f (y) =
1/c, 59 ≤ y ≤ 61;
0,
otherwise.
Answer the following questions about this random variable.
(a) What is the value of c?
(b) What is the cdf of Y ?
(c) What is the maximum possible value that Y can take?
(d) What is the expected value of Y ?
(e) What is the variance of Y ?
(f) What is the probability that Y is equal to 60?
2. Repeat the questions (b)-(f) in Problem 1 for the following pdf:
3 2 1
16 y + 4 , 0 ≤ y ≤ 2;
f (y) =
0,
otherwise.
3. A quality engineer exams a batch of manufactured parts. Each part has a probability of 0.9 to pass the exam
and 0.1 to fail the exam.
(a) What is the probability of only 1 part out of 5 parts passing the exam?
(b) What is the probability distribution of n parts out of 5 parts passing the exam? Please write down the pmf
and cdf of the distribution.
(c) What is the average number of passing parts out of every 10 parts?
4. Customers arrive at a bank teller’s drive-through window in accordance with a Poisson process, with a mean
rate of 1.2 persons per minute.
(a) What is the probability of zero arrivals in one minute?
(b) What is the probability of zero arrivals in two minutes?
(c) What is the probability of zero arrivals in the next one minute given there is no arrival in the previous minute?
(d) What is the distribution of the interarrival times?
(e) What is the probability that the interarrival time is between 2 and 3 minutes?
5. In Excel make a histogram of Xi = − ln(Ui ), for i = 1, 2, . . . , 10000, where the Ui ’s are independent Unif(0,1)
RV’s. Also answer questions below.
(a) Compute the estimate of the mean and its 95% confidence interval. Explain in words the meaning of the
confidence interval.
(b) What kind of distribution does it look like?
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