4. Survival. Let X be a positive random variable with density function f and distribution function F. Define the hazard function H(x) = -log[1 - F(x)] and the hazard rate 1 r(x) = lim P(X ≤ x + h | X >x), h40h x ≥ 0. Show that: (a) r(x) = H'(x) = f(x)/{1- F(x)}, (b) Ifr (x) increases with x then H(x)/x increases with x, (c) H(x)/x increases with x if and only if [1 - F(x)] ≤ 1- F(ax) for all 0 ≤ a ≤ 1,
4. Survival. Let X be a positive random variable with density function f and distribution function F. Define the hazard function H(x) = -log[1 - F(x)] and the hazard rate 1 r(x) = lim P(X ≤ x + h | X >x), h40h x ≥ 0. Show that: (a) r(x) = H'(x) = f(x)/{1- F(x)}, (b) Ifr (x) increases with x then H(x)/x increases with x, (c) H(x)/x increases with x if and only if [1 - F(x)] ≤ 1- F(ax) for all 0 ≤ a ≤ 1,
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![4. Survival. Let X be a positive random variable with density function f and distribution function
F. Define the hazard function H(x) = -log[1 - F(x)] and the hazard rate
1
r(x) = lim
h↓0 h
+h | X
≤x + h | X > x),
> x),
P(X ≤ x
x ≥ 0.
Show that:
(a) r(x) = H'(x) = f(x)/{1 - F(x)},
(b) Ifr (x) increases with x then H(x)/x increases with x,
(c) H(x)/x increases with x if and only if [1 - F(x)] ≤ 1-F (ax) for all 0 ≤a ≤ 1,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf7dd2ac-d180-4b71-ba19-41e287135aea%2F02a04732-c1c8-4577-a4fc-85526b0bcc39%2Fpy4mr9p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Survival. Let X be a positive random variable with density function f and distribution function
F. Define the hazard function H(x) = -log[1 - F(x)] and the hazard rate
1
r(x) = lim
h↓0 h
+h | X
≤x + h | X > x),
> x),
P(X ≤ x
x ≥ 0.
Show that:
(a) r(x) = H'(x) = f(x)/{1 - F(x)},
(b) Ifr (x) increases with x then H(x)/x increases with x,
(c) H(x)/x increases with x if and only if [1 - F(x)] ≤ 1-F (ax) for all 0 ≤a ≤ 1,
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