# Random variables and expected value

• March 2nd 2011, 11:15 PM
keityo
Random variables and expected value
Hello, I'm really stuck on this problem:

There are k types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type i with probability pi. If n coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of n coupons.)

I'm not sure how to set up this problem. I know that the random variable should be X=#of distinct types of stamps, but I am not sure where to go from there.

Thanks for the help in advance!
• March 3rd 2011, 12:10 AM
CaptainBlack
Quote:

Originally Posted by keityo
Hello, I'm really stuck on this problem:

There are k types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type i with probability pi. If n coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of n coupons.)

I'm not sure how to set up this problem. I know that the random variable should be X=#of distinct types of stamps, but I am not sure where to go from there.

Thanks for the help in advance!

Google for "multimomial distribution"

CB
• March 3rd 2011, 12:32 AM
keityo
Thank you so much! So this problem is just a multinomial distribution with expected value of npi for each coupon, and thus the expected number of different coupon is $\sum_{i=1}^{k} np_i$, right?