A student or the Kandom Processes course at lise aims to prepare for the inal 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.

$202$o Random Processes

Final Exam, Page $2$ of $3$

$02$ Dec $2024$

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?

$(3$ points$)$ Find the $($individual$)$ probability distribution of Qi$,$ for any i E k$.$ What is its expected value?

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

$(2$ 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?