The TOEFL requirement for admission at the University is 6 or more points. It wa

Question

The TOEFL requirement for admission at the University is 6 or more points. It was found that the probability that a student gets 6 or more points is 0.3. The student can repeat the test as many times as he/she wishes till they achieve the requirements. a) What is the type and the parameter(s) of the distribution that describes the random variable X, which represents the number of trials (test repetitions) till the student achieves the University requirement? Write down the pmf of X with all parameters substituted with their numeric values. b) Calculate the average number of trials (test repetitions) by any student to achieve the requirements. c) Calculate the probability that a student has to take the test 4 times till she/he achieves the requirements. d) Calculate the probability that a student will achieve the requirements at the 6th trial given that he did not meet the requirements in his first and second trials. e) Compare and Comment on part (c) and (d). f) Suppose the student needs to pay 100$ for the registration (one-time payment) and 20$ for each exam. Let Y be the random variable representing the amount he needs to pay. Find the expectation and the variance of Y.

Explanation / Answer

p = Prob of passing the test = 0.3

Let the student gets success in kth trial

P(x=k) = P(k-1 unsuccessful and k success)

= 0.7k-10.3

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c) P(x=4) = (.7^3 (.3)=0.1029

d) As independent reqd prob = p(x=6)

= (0.7)5(0.3)

e) part c is x =4 and part d is x=6

f) Y = 100+x(20)

E(X) = 1/p = 3.333

E(Y) = 100+20(10/3)

= 120.667

Var(x) = (1-p)/p^2 = 0.7/0.9 =0.7778

Var(Y) = 400(0.7778)

=311.1111

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