Question

A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is as follows.

f(x) =

kx2		0 ≤ x ≤ 2
0		otherwise

(a) Find the value of k. (Enter your answer to three decimal places.)


Draw the corresponding density curve. [Hint: Total area under the graph of f(x) is 1.]

(b) What is the probability that the lecture ends within 1 min of the end of the hour? (Enter your answer to three decimal places.)


(c) What is the probability that the lecture continues beyond the hour for between 30 and 45 sec? (Round your answer to four decimal places.)


(d) What is the probability that the lecture continues for at least 45 sec beyond the end of the hour? (Round your answer to four decimal places.)

Answer 1

(a)

For a valid pdf following must be true:

..............(1)

Now

...............(2)

Equating equation (1) and (2) gives

c= 3/8 = 0.375

Following is the curve

(b)

The probability that the lecture ends within 1 minute of the end of the hour is

(c)

30 second = 0.50 minute

45 seconds = 0.75 minute

(d)

The probability that the lecture continues for at least 45 seconds beyond the end of the hour is

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