I was practicing my probability and came across this question which i couldn't solve.
A particular professor is known for his arbitrary grading policies. Each paper receives a grade
from the set {A, A−, B+, B, B−, C+}, with equal probability, independently of other papers.
How many papers do you expect to hand in before you receive each possible grade at least once?
My first approach was to consider the random variable $X_{i}$ which denotes the number of papers before the $i^{th}$ and the $(i-1)^{th}$ grade the the total number of papers would be $X=1+\Sigma{X_{i}}$.But i cant proceed further.
Could someone please help me understand how to solve this question?
Answer
$$\mathbb E(1+X_1+\dots +X_5)=1+\mathbb EX_1+\cdots+\mathbb EX_5$$
Here $X_i$ stands for the number of papers to hand in after $i$ different gradings are received to come to $i+1$ gradings received. It is geometrically distributed with parameter $\frac{6-i}{6}$ so that $\mathbb EX_i=\frac{6}{6-i}$.
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