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C1. A collector wants to collect football stickers to fill an album. There are n unique stickers to collect. Each time the collector buys a sticker, it is one of the n stickers chosen independently uniformly at random. Unfortunately, it is likely the collector will end up having "swaps", where he has received the same sticker more than once, so he will likely need to buy more than n stickers in total to fill his album. But how many? (a) Suppose the collector has already got j unique stickers (and some number of swaps), for j= 0, 1, 2,...,n - 1. Let X; be the the number of extra stickers he buys until getting a new unique sticker. Explain why X, is geometrically distributed, and state the parameter p = p; of the geometric distribution.

C1. A collector wants to collect football stickers to fill an album. There are n unique stickers to collect. Each time the collector buys a sticker, it is one of the n stickers chosen independently uniformly at random. Unfortunately, it is likely the collector will end up having "swaps", where he has received the same sticker more than once, so he will likely need to buy more than n stickers in total to fill his album. But how many? (a) Suppose the collector has already got j unique stickers (and some number of swaps), for j= 0, 1, 2,...,n - 1. Let X; be the the number of extra stickers he buys until getting a new unique sticker. Explain why X, is geometrically distributed, and state the parameter p = p; of the geometric distribution.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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C1. A collector wants to collect football stickers to fill an album. There are n unique stickers to collect. Each time the collector buys a sticker, it is one of the n stickers chosen independently uniformly at random. Unfortunately, it is likely the collector will end up having "swaps", where he has received the same sticker more than once, so he will likely need to buy more than n stickers in total to fill his album. But how many? (a) Suppose the collector has already got j unique stickers (and some number of swaps), for • " j = 0, 1, 2,. n 1. Let X, be the the number of extra stickers he buys until getting a new unique sticker. Explain why X, is geometrically distributed, and state the parameter p = p; of the geometric distribution. (b) Hence, show that the expected number of stickers the collector must buy to fill his album is n N k=1 = (c) The World Cup 2020 sticker album required n 670 unique stickers to complete it, and stickers cost 18p each. Using the expression from (b), calculate the expected amount of money needed to fill the album. You should do this calculation in R and include the command you used in your answer. (d) By approximating the sum in part (b) by an integral, explain why the expected number of stickers required is approximately n log n, where log denotes the natural logarithm to base e.
Transcribed Image Text:C1. A collector wants to collect football stickers to fill an album. There are n unique stickers to collect. Each time the collector buys a sticker, it is one of the n stickers chosen independently uniformly at random. Unfortunately, it is likely the collector will end up having "swaps", where he has received the same sticker more than once, so he will likely need to buy more than n stickers in total to fill his album. But how many? (a) Suppose the collector has already got j unique stickers (and some number of swaps), for • " j = 0, 1, 2,. n 1. Let X, be the the number of extra stickers he buys until getting a new unique sticker. Explain why X, is geometrically distributed, and state the parameter p = p; of the geometric distribution. (b) Hence, show that the expected number of stickers the collector must buy to fill his album is n N k=1 = (c) The World Cup 2020 sticker album required n 670 unique stickers to complete it, and stickers cost 18p each. Using the expression from (b), calculate the expected amount of money needed to fill the album. You should do this calculation in R and include the command you used in your answer. (d) By approximating the sum in part (b) by an integral, explain why the expected number of stickers required is approximately n log n, where log denotes the natural logarithm to base e.
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