Joint and Marginal Distributions of a Randomly Selected Test Answer

[Kakashi]

Homework Statement

A class of n students takes a test consisting of m questions. Student i submits answers to the first mi questions only.

(a) The grader randomly selects one submitted answer from all answers that were handed in. Let
I = the student ID of the selected answer, taking values 1,2,…,n, J = the question number of the selected answer, taking values 1,2,…,m.Assume that every submitted answer is equally likely to be chosen.Determine the joint probability mass function (PMF) P(I,J). From this, compute the marginal PMFs

(b) Assume that an answer to question j if submitted by student i is correct with probability
pij. If an answer is correct, the student receives a points and b points otherwise. Calculate the expeected value of the score of student i.

Relevant Equations

Joint PMF
Marginal PMF

A class of n students takes a test with m questions. Student i submits answers to the first $$ m_{i} $$ questions. Let
$$M= \sum_{i=1}^{n} m_{i}=m_{1}+m_{2}+..m_{n} $$ denote the total number of submitted answers .
The grader randomly picks one submitted answer. We define two discrete random variables:

I (student ID) $$ 1 \leq i \leq n $$ and J (question number) $$ 1 \leq j \leq m $$. The random pair (I,J) takes values in the set $$ \{(i,j):1\leq i \leq n, 1\leq j \leq m_{i} \} $$. Since the grader chooses uniformly from all submitted answers, the joint PMF is

$$ P_{I,J} (i,j)=\begin{array}{lr} \frac{1}{M}, \text{if} \hspace{0.1cm} 1 \leq j \leq m_{i} \\ 0 , \text{if} \hspace{0.1cm} j\geq m_{i} \end{array} $$

$$ P_{I,J}=P(I=i,J=j) $$

The probability that the student picked is student i and the question picked is j is $$\frac{1}{\sum_{i=1}^{n} m_{i}} $$ if the question picked had a submitted answer and 0 otherwise.

For the marginal of I $$P_{I}=\sum_{j} P_{I,J} (i,j)={J}=\sum_{j}^{M} \frac{1}{M}=\sum_{j}^{m_{i}} \frac{1}{M}=\frac{m_{i}}{M} $$

The marginal PMF $$P_{J}=\sum_{i}^{n} P_{I,J} (i,j) $$ Union of the disjoint events $$ \{I=i, J=j\} $$ as i ranges over all the different values of i. How do we know how many students have answered question j?

b)
We define a new random variable X points awarded for a single randomly picked submitted answer. Given the picked answer is (I=i,J=j)

$$ P(X=a|(I=i,J=j))=p_{ij})$$
$$ P(X=b|(I=i,J=j))=(1-p_{ij}) $$

I am not sure how to show that the expected value of the score of student i is the sum of the expected score of the submitted questions.