# In an entrance examination a student has to answer all the $120$ questions. Each question has four options and only one option is correct. A student gets $1$ mark for a correct answer and loses half mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the answer to each question at random?

Toolbox:
• If S is a sample space with a probability measure and X is a real valued function defined over the elements of S, then X is called a random variable.
• Types of Random variables :
• (1) Discrete Random variable (2) Continuous Random variable
• Discrete Random Variable :If a random variable takes only a finite or a countable number of values, it is called a discrete random variable.
• Continuous Random Variable :A Random Variable X is said to be continuous if it can take all possible values between certain given limits. i.e., X is said to be continuous if its values cannot be put in 1 − 1 correspondence with N, the set of Natural numbers.
• The probability mass function (a discrete probability function) $P(x)$ is a function that satisfies the following properties :
• (1) $P(X=x)=P(x)=P_x$
• (2) $P(x)\geq 0$ for all real $x$
• (3) $\sum P_i=1$
• Moments of a discrete random variable :
• (i) About the origin : $\mu_r'=E(X^r)=\sum P_ix_i^{\Large r}$
• First moment : $\mu_1'=E(X)=\sum P_ix_i$
• Second moment : $\mu_2'=E(X^2)=\sum P_ix_i^2$
• (ii) About the mean : $\mu_n=E(X-\bar{X})^n=\sum (x_i-\bar{x})^nP_i$
• First moment : $\mu_1=0$
• Second moment : $\mu_2=E(X-\bar{X})^2=E(X^2)-[E(X)]^2=\mu_2'-(\mu_1')^2$
• $\mu_2=Var(X)$
Step 1:
Let $X$ be the random variable denoting the number of marks the student can score while answering 1 question.
The values $X$ can take are 1(for correct answer) or $-\large\frac{1}{2}$(for a wrong answer)
Step 2:
$P(X=1)=\large\frac{1}{4}$ (Since only one out of 4 options is correct)
$P(X=-\large\frac{1}{2})=\frac{3}{4}$
Step 3:
The probability distribution of $X$ is given by
Step 4:
$E(X)=\sum P_i x_i=1\times \large\frac{1}{4}+(-\large\frac{1}{2})\times \large\frac{3}{4}$
$\qquad\qquad\quad\;\;=\large\frac{1}{4}-\frac{3}{8}$
$\qquad\qquad\quad\;\;=\large\frac{-1}{8}$
Step 5:
There are 120 questions in the test.