3. Suppose that h(x) is the hazard function of X, and h(x)=\ for all x. (a) Calculate the survival function S(x). (b) Derive the mean life E(X) and the mean residual life E(X-x|X>x), where E(X) is a special case of mean residual life with x=0. (c) Based on your answer in (b), do you think h(x) can be used to model human lifespan?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
3. Suppose that h(x) is the hazard function of X, and h(x)=\ for all x.
(a) Calculate the survival function S(x).
(b) Derive the mean life E(X) and the mean residual life E(X-x|X>x), where E(X) is a special case of mean residual
life with x=0.
(c) Based on your answer in (b), do you think h(x) can be used to model human lifespan?
Transcribed Image Text:3. Suppose that h(x) is the hazard function of X, and h(x)=\ for all x. (a) Calculate the survival function S(x). (b) Derive the mean life E(X) and the mean residual life E(X-x|X>x), where E(X) is a special case of mean residual life with x=0. (c) Based on your answer in (b), do you think h(x) can be used to model human lifespan?
Expert Solution
Step 1: Survival function S(x)

To calculate the survival function S(x) when the hazard function h(x) is constant (i.e., h(x)=λ for all x), we can use the formula for the survival function:

S left parenthesis x right parenthesis space equals space e x p left parenthesis negative integral subscript 0 superscript x space lambda space d x right parenthesis

Since h(x)=λ for all x, we can simply integrate it with respect to x:

integral subscript 0 superscript x space lambda space d x space equals space lambda x

Now, we can plug this result into the survival function formula:

S left parenthesis x right parenthesis space equals space e x p left parenthesis negative lambda x right parenthesis

a) So, the survival function S(x) is exp(-λx).

steps

Step by step

Solved in 4 steps with 14 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,