A student aims to prepare for the final exam using a

corpus of n practice questions labeled with the integers $1,...,$ n$.$ The student unfortunately

has the bandwidth to practise only k out of the n questions. $($Note: n and k are fixed positive

integers with $1<$ k $<$ n$.)$

$($a$)(2$ points$)$ Suppose the student adopts the following strategy to choose questions to

practise. First select a question uniformly at random from all available questions.

After practising this question, discard it and select a question uniformly at random

from all remaining questions. Continue until k questions have been practised.

Let Qi be the random variable denoting the question number picked in the ith time by

the student for practising, for $1$ i $$ k$.$ Are Qi and Qj

independent for i $=$ j and

why?

$($b$)(3$ points$)$ Find the $($individual$)$ probability distribution of Qi

$,$ for any i in $[$k$].$ What

is its expected value?

$($c$)(2$ points$)$ What is the expected value of the total number of times the student practices question number $1?$

$($d$)(2$ points$)$ Suppose the student adopts the following strategy to choose questions to

practice. First select a question uniformly at random from all available questions.

After practising this question, put it back in the pool of all available questions. Select a

question uniformly at random from all available questions. Continue until k questions

have been practised.

Let Ri be the random variable denoting the question number picked in the ith time by

the student for practising, for $1$ i $$ k$.$ Are Ri and Rj

independent for i $=$ j and

why?

$($e$)(3$ points$)$ Find the $($individual$)$ probability distribution of Ri

$,$ for any i$.$ What is its

expected value?

$($f$)(3$ points$)$ What is the expected value of the total number of times the student practices question number $1?[$Hint: A neat way is to write this random variable in terms

of indicator random variables based on R$1,$ R$2,...,$ Rk$,$ and then use a property of the

expected value.$]$