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A function sometimes associated with continuous
Chapter 4, Problem 190SE(choose chapter or problem)
A function sometimes associated with continuous nonnegative random variables is the failure
rate (or hazard rate) function, which is defined by
\(r(t)=\frac{f(t)}{1-F(t)}\)
for a density function \(f(t)\) with corresponding distribution function \(F(t)\). If we think of the
random variable in question as being the length of life of a component, \(r(t)\) is proportional to
the probability of failure in a small interval after 𝑡, given that the component has survived up
to time 𝑡. Show that,
a for an exponential density function, \(r(t)\) is constant.
b for a Weibull density function with \(m>1\), \(r(t)\) is an increasing function of 𝑡. (See Exercise 4.186.)
Equation Transcription:
Text Transcription:
r(t)=f(t) over 1-F(t)
f(t)
F(t)
r(t)
r(t)
m>1
r(t)
Questions & Answers
QUESTION:
A function sometimes associated with continuous nonnegative random variables is the failure
rate (or hazard rate) function, which is defined by
\(r(t)=\frac{f(t)}{1-F(t)}\)
for a density function \(f(t)\) with corresponding distribution function \(F(t)\). If we think of the
random variable in question as being the length of life of a component, \(r(t)\) is proportional to
the probability of failure in a small interval after 𝑡, given that the component has survived up
to time 𝑡. Show that,
a for an exponential density function, \(r(t)\) is constant.
b for a Weibull density function with \(m>1\), \(r(t)\) is an increasing function of 𝑡. (See Exercise 4.186.)
Equation Transcription:
Text Transcription:
r(t)=f(t) over 1-F(t)
f(t)
F(t)
r(t)
r(t)
m>1
r(t)
ANSWER:
Solution 190SE
Step1 of 3:
Let us consider a Hazard rate failure rate of function:
Where,
Density function.
distribution function.
Here our goal is:
a). We need to show that for an exponential density function, r (t) is constant.
b). We need to show that for a Weibull density function with m > 1, r (t) is an increasing function of t.
Step2 of 3:
a).
We know that the density function of exponential distribution is:
Similarly,
The distribution function of exponential distribution is: