Coupon problem from Ross' Probability Models

Asked May 14 '20 at 10:18

Active May 14 '20 at 15:18

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This is a question from Sheldon M Ross' Intro to Probability Models.

Suppose that each coupon obtained is, independent of what has been previously obtained, equally likely to be any of $m$ different types. Find the expected number of coupons one needs to obtain in order to have at least one of each type.

Hint: Let $X$ be the number needed. It is useful to represent $X$ by $X = \sum_{i=1}{X_i}$ where each $X_i$ is a geometric random variable.

I did it using this hint and I now get $m^2$ as $E[X]$. I'm not sure if this is right and more importantly, I'm not able to understand why/how it can be modeled as the sum of many geometric distributions.

My initial approach was to define the indicator random variable

$$X_i = \begin{cases} 1 & \textrm{if ith coupon is new} \\ 0 & \textrm{otherwise} \end{cases} $$

and then take $X = \sum_{i=1}{X_i}$ but I'm not sure how to proceed with that.

edited May 14 '20 at 15:18

joriki

242,601

asked May 14 '20 at 10:18

newtothis

Hint: Let $X_i$ be the time to get the $i^{\text{th}}$ new coupon after you have obtained $i - 1$ different kinds. – sudeep5221 May 14 '20 at 12:06
Oh. But do I then consider time in some discrete steps?
Also, is $m^2^ the right answer?
– newtothis May 14 '20 at 13:07
Yes you will have to consider time in discrete steps. And the answer is $m \log m$. – sudeep5221 May 14 '20 at 17:34

Coupon problem from Ross' Probability Models

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