You are taking a multiple-choice test with n questions each of which has 4 alternatives. You have mastered 60% of the material

Question

You are taking a multiple-choice test with n questions each of which has 4 alternatives. You have mastered 60% of the material. Assume this means that you have a 0.6 chance of knowing the answer to a random test question, and that if you don’t know the answer to a question then you randomly select among the four answer choices. Assume that this holds for each question, independent of the others, and assume that each correct answer gives 1 point and wrong answers give 0 points, the score is the sum of all points. For each answer define a random variable Xi (i=1,2,...,n) that takes the value 1 if the ith answer is correct and 0 otherwise. a.What is the probability that you answer a particular question correctly? b.What is your expected score on the exam? c.Write down a formula for the probability mass function (pmf) for one particular X, obtain the cumulative distribution function (CDF) for Xi and plot the CDF WORK: for my work so far I have A = Knowing the answer B = All choices are equal and C = Student answers correctly. P(A) = .6, P(B) = .25 I am looking for P(A|C)? = P(C|A)P(A)/P(C)? Other than that I am kind of lost

Answer 1

a) The probability that he gets a particular question correct is

$$p=\left(0.6\cdot1\right)+\left(0.4\cdot0.25\right)=0.7$$

b) This is a binomial distribution with mean $np$

c)

Assuming $X$ is a random variable taking on the value $1$ if the answer is correct and $0$ if the answer is incorrect...

$$ p_{X}(x)= \begin{cases} 0.7 & x =1 \\ 0.3 & x=0 \\ 0 & \text{otherwise} \end{cases} $$

d)

$$ F_{X}(x)= \begin{cases} 1 & x \geq 1 \\ 0.3 & 0\leq x\lt 1 \\ 0 & x \lt 0 \end{cases} $$

Note that even in the discrete case, we must account for all $x\in\mathbb{R}$