Question

A spacecraft is fitted with four radios. The expected lifetime of each radio can be modelled using an exponential distribution with a mean of 5 months. All four radios are used from the beginning of the mission to maximise bandwidth. The spacecraft will be unusable once all four radios fail. What will be the lifetime of the spacecraft in the first realisation if the first four numbers generated by the random number generator are 0.36, 0.422, 0.174 and 0.279? (Answer to the nearest month).

Answer 1

Let X be the lifetime of a randomly choosen radio. X has an Exponential distribution with mean

The cdf of X is

We will use inverse transformation method to generate 4 instances of the exponential random variable.

Let U be Uniformly distribution in (0,1)

We will find the inverse of F(X)

the steps for generating the random variable X are

• Draw a random number u from uniform (0,1)
• is a draw of X

Now we generate 4 random numbers from X

1. u=0.36,
2. u=0.422,
3. u=0.174,
4. u=0.279,

The lifetime of 4 radios would be 2,3,1, and 2 months respectively.

The lifetime of the spacecraft would be 3 months (in this realization)