A test has two questions. Let (X, Y) be the outcomes for a randomly selected stu

Question

A test has two questions. Let (X, Y) be the outcomes for a randomly selected student. X = 1 if the student answers Question 1 correctly; and X = 0 if the student does not. Y = 1 if the student answers Question 2 correctly; and Y = 0 if the student does not. The joint pmf of (X, Y) is given below.

Marginal dsitribution of Y

a.) What is the probability that a student answers both questions correctly?

b.) If the student answered Question 1 correctly, what is the probability that he also answered Question 2 correctly?

c.) From the marginal distribution, you can get the following: E(X) = 0.5 and Var(X) = 0.25. What is the correlation between X and Y?

d.) The test scores are determined as follows: the student gets 50 points for answering a question corrrectly and zero otherwise. There are no partial credits. Let S be the total score that the student will get in the test. What are the mean and variance of S? (Do not compute this from the distribution of S. Use your answers in parts (a) and (b) and the other information you are given to answer the question.)

e.) Write down the pmf of S.

y 0 1 Marginal distribution of X x 0 0.4 0.1 0.5 1 0.1 0.4 0.5

Marginal dsitribution of Y

0.5 0.5 1.0

Explanation / Answer

a) probability that a student answers both questions correctly =P(x=1,y=1) =0.4

b)probability that he also answered Question 2 correctly given probability that he also answered Question 2 correctly

=P(Y=1|X=1) =0.4/0.5=4/5=0.8

c)here E(X)=E(Y) =0.5

Var(X)=Var(Y)=0.25

E(XY)=0.4

therefore covar(X,Y)=E(XY)-E(X)E(Y)=0.4-0.5*0.5=0.15

hence correlation between X and Y =Covar(X,Y)/(Var(X)*Var(Y))1/2 =0.6

d)total score S =50(X+Y)

mean of S =E(S)=50*(E(X)+E(Y))=50*(0.5+0.5)=50

Variance of S =Var(S)=502*(Var(X)+Var(Y)+2*Covar(X,Y)) =2500*(0.25+0.25+2*0.15)=2500*0.8=2000

e) from above pmf of S is as follows:

P(S=0)=0.4

P(S=50)=0.2

P(S=100)=0.4

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