I need help with parts D and E please. 2. A college professor never finishes his lecture before the end of the hour and alwaysfinishes his lecture within 2 min after the hour. Let X be the time that elapses betweenthe end of the hour and the end of the lecture and suppose the pdf of X is f(x) = kx²for 0

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Description: I need help with parts D and E please. 2. A college professor never finishes his lecture before the end of the hour and alwaysfinishes his lecture within 2 min after the hour. Let X be the time that elapses betweenthe end of the hour and the end of t

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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I need help with parts D and E please.

2. A college professor never finishes his lecture before the end of the hour and always
finishes his lecture within 2 min after the hour. Let X be the time that elapses between
the end of the hour and the end of the lecture and suppose the pdf of X is f(x) = kx²
for 0 <x<2 and f(x) = 0 elsewhere.
(a) Find the value of k.
(b) What is the probability that the lecture ends within 1 min of the end of the hour?
(c) What is the probability that the lecture continues beyond the hour for between 60
and 90 sec?
(d) What is the probability that the lecture continues for at least 90 sec beyond the end
of the hour?
(e) Find the cdf of X, F(x), and repeat parts (b), (c), and (d) using F(x) instead of f (x).

Expert Solution

Step 1

Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and specify the other subparts

Part A.

f(x) always integrate to 1.

$\int f (x) d x = 1$

Plug the value

$\int_{0}^{2} k x^{2} d x = 1$

Take constant value outside

$k \cdot \int_{0}^{2} x^{2} d x = 1$

Using Power rule we have :

$k {[\frac{x^{2 + 1}}{2 + 1}]}_{0}^{2} = 1$

$k {[\frac{x^{3}}{3}]}_{0}^{2} = 1$

$k [\frac{2^{3}}{3} - \frac{0^{3}}{3}] = 1$

$k [\frac{8}{3} - \frac{0}{3}] = 1$

$k [\frac{8}{3}] = 1$

$k = \frac{3}{8}$

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2. A college professor never finishes his lecture before the end of the hour and always finishes his lecture within 2 min after the hour. Let X be the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is f(x) = kx² for 0