Question

Give a method for simulating a random variable having failure rate function (a) λ(t) = c;

(b) λ(t) = ct;

(c) λ(t) = ct2;

(d) λ(t) = ct3.

Short answer

We have used use the university of the uniform in order to obtain the required distribution of all subparts of the question.

Step-by-step solution

Part (a) Step 1: Given Information

We have to find method for simulating a random variable having failure rate function$\lambda \left(t\right)=c$.

Part (a) Step 2: Explanation

Using the relation between Hazzard rate and CDF

$F\left(t\right)=1-\text{exp}\left(-{\int }_0^t\lambda \left(s\right)ds\right)$

Consider$\lambda \left(t\right)=c$that means

$F\left(t\right)=1-e^{-ct}$

Part (b) Step 1: Given Information

We have to find method for simulating a random variable having failure rate function$\lambda \left(t\right)=ct$.

Part (b) Step 2: Explaantion

Consider $\lambda \left(t\right)=ct$which means

$F\left(t\right)=1-e^{-c\frac{t^2}{2}}$

Generating$U\in \left(0,1\right)$randomly and declare $X={\left(-\frac{2}{c}\text{log}\left(1-U\right)\right)}^{1/2}$. By the universality of the Uniform, we have$X$follows required distribution.

Part (c) Step 1: Given Information

We have to find method for simulating a random variable having failure rate function$\lambda \left(t\right)=c{t}^2$.

Part (c) Step 2: Explanation

Consider $\lambda \left(t\right)=c{t}^2$which implies

$F\left(t\right)=1-e^{-c\frac{t^3}{3}}$

Generating$U\in \left(0,1\right)$randomly and declare $X={\left(-\frac{3}{c}\text{log}\left(1-U\right)\right)}^{1/3}$. By the universality of the Uniform, we have that $X$follows required distribution.

Part (d) Step 1: Given Information

We have to find method for simulating a random variable having failure rate function$\lambda \left(t\right)=c{t}^3$.

Part (d) Step 2: Simplify

Consider$\lambda \left(t\right)=c{t}^3$which implies

$F\left(t\right)=1-e^{-c\frac{t^4}{4}}$

Generating $U\in \left(0,1\right)$randomly and declare $X={\left(-\frac{4}{c}\text{log}\left(1-U\right)\right)}^{1/4}$. By the universality of the Uniform, we have that $X$follows required distribution.