Question

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) = kx^2 0 ? x ? 2
(a) Find the value of k. (Enter your answer to three decimal places.)
(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 15 and 45 sec? (Round your answer to four decimal places.)
(d) What is the probability that the lecture continues for at least 75 sec beyond the end of the hour? (Round your answer to four decimal places.)

Answer 1

To solve the student's series of probability questions, the constant 'k' is first determined by integrating the provided pdf over the interval [0, 2]. Following this, the probabilities for the various time intervals are calculated using the determined pdf.

Step-by-step explanation:

The student's problem involves finding the constant k and calculating probabilities for a continuous random variable X with varying constraints, based on a given probability density function (pdf). The pdf is f(x) = kx2 over the interval 0 ≤ x ≤ 2. To find k, we ensure the pdf integrates to 1 over the specified interval:

1. Integrate f(x) over the interval [0, 2] such that ∫02 kx2 dx = 1.
2. Calculate the resultant integral to find the value of k.
3. Use this value of k and integrate f(x) over the desired intervals to find the respective probabilities.

For instance, the probability that the lecture ends within 1 minute of the hour is found by integrating f(x) from 0 to 1, and so on for other requested probabilities in the problem.

Answer 2

To find the value of k, integrate the probability density function (pdf) over its entire range and equate it to 1. The value of k is 0.375. The probability that the lecture ends within 1 min of the hour is 0.125.

Step-by-step explanation:

(a) To find the value of k, we need to calculate the integral of the probability density function (pdf) over its entire range. The integral of f(x) from 0 to 2 should equal 1, since the total probability in a continuous distribution must be 1. So we have:

∫02kx2dx = 1

Integrating kx2 with respect to x gives:

k[(x3/3)]02 = 1

k(8/3 - 0) = 1

k = 3/8 = 0.375

(b) To find the probability that the lecture ends within 1 min of the end of the hour, we need to calculate the integral of f(x) from 0 to 1. This represents the area under the curve within this range:

P(0 ≤ X ≤ 1) = ∫01(0.375x2)dx

P(0 ≤ X ≤ 1) = (0.375/3)x3|01

P(0 ≤ X ≤ 1) = (0.375/3)(13-03)

P(0 ≤ X ≤ 1) = 0.375/3 = 0.125