John sells T-shirts at the beach. There are two types of shirts: priced $10 and

Question

John sells T-shirts at the beach. There are two types of shirts: priced $10 and $15. People stop at the stand according to a Poisson process with rate = 10 per hour. However, each customer buys a shirt with probability p = 0.3, and if she/he buys, a cheaper shirt is bought with probability 0.75. Nobody buys two shirts. (a) Does the number of people who bought shirts have a Poisson distribution? If no, give an example, if yes, find the intensity of the corresponding Poisson process. (b) Does the number of people who bought shirts for $10 have a Poisson distribution? (c) Are the numbers of people who bought shirts for $10 and for $15 independent r.v.? (d) Findtheprobabilitythatthefirstshirtwillbepurchasedduringthefirsthour,andbefore this purchase two customers will stop at the stand but will not buy anything. (Advice: This means that there will be at least three arrivals during the first hour but the first two arrivals will be unlucky for John. Look up the joint distribution of N1,...,Nl given N = n in Section 3.3.2.1.) (e) Using normal approximation, estimate the probability that the total sales of shirts dur- ing 8 hours is greater than $250. John sells T-shirts at the beach. There are two types of shirts: priced $10 and $15. People stop at the stand according to a Poisson process with rate = 10 per hour. However, each customer buys a shirt with probability p = 0.3, and if she/he buys, a cheaper shirt is bought with probability 0.75. Nobody buys two shirts. (a) Does the number of people who bought shirts have a Poisson distribution? If no, give an example, if yes, find the intensity of the corresponding Poisson process. (b) Does the number of people who bought shirts for $10 have a Poisson distribution? (c) Are the numbers of people who bought shirts for $10 and for $15 independent r.v.? (d) Findtheprobabilitythatthefirstshirtwillbepurchasedduringthefirsthour,andbefore this purchase two customers will stop at the stand but will not buy anything. (Advice: This means that there will be at least three arrivals during the first hour but the first two arrivals will be unlucky for John. Look up the joint distribution of N1,...,Nl given N = n in Section 3.3.2.1.) (e) Using normal approximation, estimate the probability that the total sales of shirts dur- ing 8 hours is greater than $250. John sells T-shirts at the beach. There are two types of shirts: priced $10 and $15. People stop at the stand according to a Poisson process with rate = 10 per hour. However, each customer buys a shirt with probability p = 0.3, and if she/he buys, a cheaper shirt is bought with probability 0.75. Nobody buys two shirts. (a) Does the number of people who bought shirts have a Poisson distribution? If no, give an example, if yes, find the intensity of the corresponding Poisson process. (b) Does the number of people who bought shirts for $10 have a Poisson distribution? (c) Are the numbers of people who bought shirts for $10 and for $15 independent r.v.? (d) Findtheprobabilitythatthefirstshirtwillbepurchasedduringthefirsthour,andbefore this purchase two customers will stop at the stand but will not buy anything. (Advice: This means that there will be at least three arrivals during the first hour but the first two arrivals will be unlucky for John. Look up the joint distribution of N1,...,Nl given N = n in Section 3.3.2.1.) (e) Using normal approximation, estimate the probability that the total sales of shirts dur- ing 8 hours is greater than $250.

Explanation / Answer

Part (a)

Answer: No. Number of people who bought shirts DO NOT have a Poisson distribution. It has a Binomial distribution. Because,

Suppose X = number of people who stop at the stand. Not all of them will buy a shirt. i.e., each of these X people will either buy or will not buy => there are two and only two possibilities. So, number of people who buy shirts is Binomial. Further, given probability of buying is 0.3, if Y = number of people who buy shirts, then Y ~ B(X, 0.3). ANSWER

Part (b)

Number of people who bought shirts for $10 DO NOT have a Poisson distribution by the very logic as in Part (a). Further, given probability of buying is 0.3 and if buying, probability of buying $10 shirt is 0.75, probability of buying $10 shirt is (0.3 x 0.75) = 0.225. So, if Z = number of people who buy $10 shirts, then Z ~ B(X, 0.225). ANSWER

Part (c)

Of the X people stopping at the stand, Y will buy. Of these Y, Z will buy $10 shirt and so (Y - Z) = W, say will buy $15 shirt.[Note that nobody buys two shirts.]. So, W depends on Z. Hence these random variables cannot be independent. ANSWER

Part (d)

The first shirt will be purchased during the first hour, and before this purchase, two customers will stop at the stand but will not buy anything => 3 customers stop at the stand in one hour, two do not buy and one buys.

So, P(first shirt will be purchased during the first hour, and before this purchase, two customers will stop at the stand but will not buy anything)

= P(3 customers stop at the stand in one hour) x P(two do not buy) x P(one buys)

P(3 customers stop at the stand in one hour) = P(X = 3) = 0.007567 [using Excel Function for Poisson with = 10 and x = 3].

P(two do not buy) = 0.7 x 0.7 [given P(buying) is 0.3, P(not buying) is 0.7 ]

P(one buys) = 0.3 [given]

Thus, required probability = 0.007567 x 0.49 x 0.3 = 0.0011 ANSWER

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