Exercise 2 1E[W] = N(1 + ++21N-1+RemarkNote 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 numberof caps grows with order Nlog(N); e.g. twice the number of bottle caps meansthe average number of sodas needed for a complete set more than doubles, butit's not as bad as four times as many sodas.(18)We will take the "Bottle Caps Problem" to mean, rather vaguely, analyzingeverything we can about the distribution of the number of sodas one must buyto 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 ofsoda needed to obtain a complete set of N different bottle caps.HintNote 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 ProblemAs a product promotion (with a patriotic theme) for the month of July, a softdrink manufacturer produces glass bottles of soda with the old-fashioned metalcap. Printed on the inside of each bottle cap, is the name of a US state orterritory, and a scene from that state or territory. Presumably, each state orterritory 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 PuertoRico, the District of Columbia, the US Virgin Islands, Guam, and AmericanSamoa), one can turn the set in and win a nifty prize. On average, how manybottles of soda does it take it accumulate a complete set? (Assume there's notrading with friends! We'll save that for later...)SolutionWe'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 havingi-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 + + XNAnd 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 ofN.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. Soby Equation (14), E[X₂]Continuing this reasoning with the third,fourth, etc.successive new bottle cap, the expected number of sodas one mustbuy to get a complete set of N different bottle caps is:

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Exercise 2 1E[W] = N(1 + ++21N-1+RemarkNote 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 numberof caps grows with order Nlog(N); e.g. twice the number of bottle caps meansthe average number of sodas needed for a complete set more than doubles, butit's not as bad as four times as many sodas.(18)We will take the &#34;Bottle Caps Problem&#34; to mean, rather vaguely, analyzingeverything we can about the distribution of the number of sodas one must buyto 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 ofsoda needed to obtain a complete set of N different bottle caps.HintNote 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 ProblemAs a product promotion (with a patriotic theme) for the month of July, a softdrink manufacturer produces glass bottles of soda with the old-fashioned metalcap. Printed on the inside of each bottle cap, is the name of a US state orterritory, and a scene from that state or territory. Presumably, each state orterritory 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 PuertoRico, the District of Columbia, the US Virgin Islands, Guam, and AmericanSamoa), one can turn the set in and win a nifty prize. On average, how manybottles of soda does it take it accumulate a complete set? (Assume there's notrading with friends! We'll save that for later...)SolutionWe'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 havingi-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 + + XNAnd 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 ofN.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. Soby Equation (14), E[X₂]Continuing this reasoning with the third,fourth, etc.successive new bottle cap, the expected number of sodas one mustbuy to get a complete set of N different bottle caps is:

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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.