Solution: 2019 Winter Final - 24
Author: Michiel SmidQuestion
Let $n \geq 2$ be an integer, and let $p$ and $q$ be real numbers with $0 < p < 1$ and $0 < q < 1$.
Consider $n$ students $S_1,S_2,...,S_n$ who hand in an assignment for COMP 2804. This assignment has 2 questions. For $i = 1,2,...,n$ let $A_i$ be the assignment handed in by student $S_i$.
Alexa and Zoltan are TAs for the course. Alexa is supposed to mark the first question, whereas Zoltan is supposed to mark the second question. Since both TAs are lazy, they use the following scheme (all random choices are mutually independent):
For $i = 1,2,...,n$,
Hint: Use indicator random variables.
Consider $n$ students $S_1,S_2,...,S_n$ who hand in an assignment for COMP 2804. This assignment has 2 questions. For $i = 1,2,...,n$ let $A_i$ be the assignment handed in by student $S_i$.
Alexa and Zoltan are TAs for the course. Alexa is supposed to mark the first question, whereas Zoltan is supposed to mark the second question. Since both TAs are lazy, they use the following scheme (all random choices are mutually independent):
For $i = 1,2,...,n$,
- Alexa marks the first question of assignment $A_i$ with probability $p,$
- Zoltan marks the second question of assignment $A_i$ with probability $q.$
- X = the number of assignments that are marked by both Alexa and Zoltan.
Hint: Use indicator random variables.
(a)
$(p + q) \cdot n$
(b)
None of the above.
(c)
$p \cdot q \cdot n$
(d)
$\frac{n}{p \cdot q}$
Solution
-
Determine indicator random variable
Let $X_i$ be 1 if both questions are marked by Alexa and Zoltan, so that an assignment has been marked by both Alexa and Zoltan
The probability that Alexa marks the first question is p
The probability that Zoltan marks the first question is q
$ \Pr(X_i=1) = p \cdot q $
- Profit
$ \mathbb{E}(X) = \sum_{i=1}^{n} \mathbb{E}(X_i) $
$ \mathbb{E}(X) = \sum_{i=1}^{n} p \cdot q $
$ \mathbb{E}(X) = n \cdot p \cdot q $