Exercise 2 Find a finite sum giving the variance of the number of bottles of soda needed to obtain a complete set of N different bottle caps.
Exercise 2 Find a finite sum giving the variance of the number of bottles of soda needed to obtain a complete set of N different bottle caps.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Exercise 2
![1
E[W] = N(1 + ++
2
1
N-1
+
Remark
Note that this expression implies the rather interesting conclusion that the ex-
pected number of sodas to obtain a complete set of caps grows as the number
of caps grows with order Nlog(N); e.g. twice the number of bottle caps means
the average number of sodas needed for a complete set more than doubles, but
it's not as bad as four times as many sodas.
(18)
We will take the "Bottle Caps Problem" to mean, rather vaguely, analyzing
everything we can about the distribution of the number of sodas one must buy
to obtain a complete set of bottle caps. Having done the mean (expected value),
another item of interest is the variance.
Then use Equation (16).
Exercise 2 Find a finite sum giving the variance of the number of bottles of
soda needed to obtain a complete set of N different bottle caps.
Hint
Note the X₂'s are independent (think about it), so the variances add:
Var [W] = Var[X₁] + Var[X₂] ++ Var[XN]
(19)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F639bae3a-5ded-48bd-8b36-7a1e93cf97a2%2Fns9q5i3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1
E[W] = N(1 + ++
2
1
N-1
+
Remark
Note that this expression implies the rather interesting conclusion that the ex-
pected number of sodas to obtain a complete set of caps grows as the number
of caps grows with order Nlog(N); e.g. twice the number of bottle caps means
the average number of sodas needed for a complete set more than doubles, but
it's not as bad as four times as many sodas.
(18)
We will take the "Bottle Caps Problem" to mean, rather vaguely, analyzing
everything we can about the distribution of the number of sodas one must buy
to obtain a complete set of bottle caps. Having done the mean (expected value),
another item of interest is the variance.
Then use Equation (16).
Exercise 2 Find a finite sum giving the variance of the number of bottles of
soda needed to obtain a complete set of N different bottle caps.
Hint
Note the X₂'s are independent (think about it), so the variances add:
Var [W] = Var[X₁] + Var[X₂] ++ Var[XN]
(19)
![3 The Bottle Cap Problem
As a product promotion (with a patriotic theme) for the month of July, a soft
drink manufacturer produces glass bottles of soda with the old-fashioned metal
cap. Printed on the inside of each bottle cap, is the name of a US state or
territory, and a scene from that state or territory. Presumably, each state or
territory appears at random with an equal probability. If one manages to ac-
cumulate a complete set of all 55 different bottle caps (50 states, plus Puerto
Rico, the District of Columbia, the US Virgin Islands, Guam, and American
Samoa), one can turn the set in and win a nifty prize. On average, how many
bottles of soda does it take it accumulate a complete set? (Assume there's no
trading with friends! We'll save that for later...)
Solution
We'll do the problem for the general case of N different bottle caps.
Let the random variable X, be the number of sodas it takes to get from having
i-1 different bottle caps to i different bottle caps. Then the number of sodas,
W, one needs to get a complete set of bottle caps is:
W X₁ + X₂ + + XN
And hence the expected value we are interested in is:
E[W] = E[X₁] + E[X₂] ++ E[XN]
(17)
The random variable X₁ is degenrate. When our hypothetical little boy (or girl)
starts out (zero sodas consumed ), with probability 1 = the first bottle of
N.
soda he buys will be one he doesn't have already! So trivially, E[X₁] = 1 =
=
Observe that X2 is a geometric random variable with parameter p= ¹. So
by Equation (14), E[X₂]
Continuing this reasoning with the third,
fourth, etc.successive new bottle cap, the expected number of sodas one must
buy to get a complete set of N different bottle caps is:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F639bae3a-5ded-48bd-8b36-7a1e93cf97a2%2Fx080x6h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3 The Bottle Cap Problem
As a product promotion (with a patriotic theme) for the month of July, a soft
drink manufacturer produces glass bottles of soda with the old-fashioned metal
cap. Printed on the inside of each bottle cap, is the name of a US state or
territory, and a scene from that state or territory. Presumably, each state or
territory appears at random with an equal probability. If one manages to ac-
cumulate a complete set of all 55 different bottle caps (50 states, plus Puerto
Rico, the District of Columbia, the US Virgin Islands, Guam, and American
Samoa), one can turn the set in and win a nifty prize. On average, how many
bottles of soda does it take it accumulate a complete set? (Assume there's no
trading with friends! We'll save that for later...)
Solution
We'll do the problem for the general case of N different bottle caps.
Let the random variable X, be the number of sodas it takes to get from having
i-1 different bottle caps to i different bottle caps. Then the number of sodas,
W, one needs to get a complete set of bottle caps is:
W X₁ + X₂ + + XN
And hence the expected value we are interested in is:
E[W] = E[X₁] + E[X₂] ++ E[XN]
(17)
The random variable X₁ is degenrate. When our hypothetical little boy (or girl)
starts out (zero sodas consumed ), with probability 1 = the first bottle of
N.
soda he buys will be one he doesn't have already! So trivially, E[X₁] = 1 =
=
Observe that X2 is a geometric random variable with parameter p= ¹. So
by Equation (14), E[X₂]
Continuing this reasoning with the third,
fourth, etc.successive new bottle cap, the expected number of sodas one must
buy to get a complete set of N different bottle caps is:
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