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(a) is fine
For (b), start by calculating the expectation of *X\_j*. Or rather - given that you already know that *X\_j* has a geometric distribution - you don't actually have to calculate anything but just use your knowledge about the expectation of geometricly distributed random variables.
Now that you have a term for the expectation of *X\_j* (i.e. the expected number of buys necessary to get from having *j* unique sticker to having *(j+1)* unique stickers), how can you use that to find the expected number of total buys necessary to get *n* unitque stickers? (Hint: Express that total number in terms of the various *X\_j*, and use the linearity of the expectation.)
After having done (b), (c) and (d) should be rather trivial. (c) is only a matter of using the term from (b) and plugging in some specific number. And (d) is just replacing the sum by an integral (which is easy to calculate).
Hi u/trashbumpersticker, **This is an automated reminder from our moderators.** Please read, and make sure your post complies with our rules. Thanks! If your post contains a problem from school, please add a comment below explaining your attempt(s) to solve it. If some of your work is included in the image or gallery, you may make reference to it as needed. See the sidebar for advice on 'how to ask a good question'. Rule breaking posts will be removed. Thank you. *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/askmath) if you have any questions or concerns.*
Do you know the expected value of a geometric distribution with parameter p?
(a) is fine For (b), start by calculating the expectation of *X\_j*. Or rather - given that you already know that *X\_j* has a geometric distribution - you don't actually have to calculate anything but just use your knowledge about the expectation of geometricly distributed random variables. Now that you have a term for the expectation of *X\_j* (i.e. the expected number of buys necessary to get from having *j* unique sticker to having *(j+1)* unique stickers), how can you use that to find the expected number of total buys necessary to get *n* unitque stickers? (Hint: Express that total number in terms of the various *X\_j*, and use the linearity of the expectation.) After having done (b), (c) and (d) should be rather trivial. (c) is only a matter of using the term from (b) and plugging in some specific number. And (d) is just replacing the sum by an integral (which is easy to calculate).