Question

There is an study which contains 4 questions as follow:

I) 3 questions have four multiple choices a, b, c and d

II) only one question is true and false

Let \( X \) denotes the number of correct answers for part (I) and \( Y \) denotes the number of correct answers in true/false part. Find the joint probability distribution function \( f_X,_Y(x,y) \)

Answer 1

Given

Total number of questions = 4

Part I: 3 questions with four multiple choices

Part II: One true/false question

X = Number of correct answers for Part I

Y = Number of correct answers in true/false part (Part II)

To determine: Joint probability function

Solution

X has a Binomial distribution with success probability p = 1/4 and n = 3. This is because each question can be considered as a bernoulli trial with success probability p = 1/4 and there are three questions in part I.

Hence, the probability distribution of X is:

(Binomial distribution)

   for x = 0,1,2,3 (Substituting the values of p and n)

  

  

  

  

Similarly Y has Bernoulli distribution with success probability p = 1/2.

Hence, the probability distribution of X is:

(Binomial distribution)

   for y = 0,1 (Substituting the value of p)

  

  

  

  

The outcome of part II does not depend on that of part I and vice versa. Hence, X and Y are independent.

Hence, the joint probability distributionof X and Y is:

for x = 0,1,2,3 and y = 0,1

Now tabulating the joint probability for different values of X and Y:

	X
	0	1	2	3
Y	0
1