Minimum of Exponential Random Variables: Monte Carlo Inversion Sampling

Question

Could anyone please help me one the following question?

10 phones are fully charged, the battery lifetime of any phone follows an exponential distribution with a mean of 2 hours.

(1) what is the distribution of the time until the first phone is dead?

(2) Derive an inversion sampling algorithm to sample from this distribution.

I got stuck at the first part and didn’t know how to solve this problem under the mathematical domain. But I think once I get the answer for the first part, the following part shouldn’t be a problem.

Any help is highly appreciated.

Answer 1

You want the minimum of ten independent exponential random variables. Here are three approaches to understanding the distribution of their minimum:

Intuitive. An intuitive approach is to say that failure rate is $\lambda = 1/\mu = 1/2$ or one phone every two hours. The overall rate for the 'system' of ten phones is $10\lambda = 5$ per hour. So the time until the first failure (first 'damage' to the system) is $V = \min(X_1, \dots, X_{10})\,$ with $V \sim \mathsf{Exp}(\lambda=5)\equiv \mathsf{Exp}(\mu = 0.2).$

Analytic. A formal approach is to find the CDF $F_V(t)$ of $V$. It is easier to find the reliability function $R_V(t) = 1 - F_V(t),$ for $t > 0,$ as follows: $$R_V(t) = P(V > t) = P(X_1 > t,\, X_2 > t, \dots, X_{10} > t)\\ = \prod_{i=1}^{10} P(X_i > t) = \prod_{i=1}^{10} e^{-.5t} = (e^{-.5t})^{10} = e^{-5t},$$ which implies that $F_V(t) = 1 - e^{-5t}$ for $t > 0,$ and thus $V \sim \mathsf{Exp}(\lambda=5)$ with $E(V) = 1/5 = 0.2.$

Simulation. A simulation approach begins by figuring out how to use a randomly generated realization $U \sim \mathsf{Unif}(0,1)$ to get $Y \sim \mathsf{Exp}(\lambda = 1).$ I suppose your text explains that you can use the inverse CDF (or quantile) function of $Y$ to do this. You seem to imply that you have already done this part to find that $X = -2\log(U) \sim \mathsf{Exp}(\mu = 2).$

Then you can use a program such as the one below to simulate the distribution of $V.$ In R statistical software the simulation can be programmed as follows:

set.seed(1803)             # retain seed for exactly same sumulation
m = 10^5;  n = 10;  u = runif(m*n)
MAT.U = matrix(u, nrow=m)  # m x n matrix of U's
MAT.X = -2*log(MAT.U)      # m x n matrix of X's; ten phones per row
v = apply(MAT.X, 1, min)   # m-matrix of V's  (minimums of rows)
mean(v)
## 0.1996706               # aprx E(V) = 0.2

hist(v, prob=T, br=50, ylim=c(0,5), col="skyblue2", main="Simulated Dist'n of Minimums")
  curve(dexp(x, rate=5), add=T, col="maroon", n=1001, lwd=2)

The figure shows the simulated distribution of $V$ (based on $m = 100,000$ simulated observations), with the density curve of $\mathsf{Exp}(\mu=0.2)$ superimposed (maroon curve).