Page 446 - A First Course In Stochastic Models

B. USEFUL PROBABILITY DISTRIBUTIONS 441

In other words, the minimum of the two exponentially distributed lifetimes X 1 and
X 2 is exponentially distributed with mean 1/(λ 1 + λ 2 ) and the probability that the
lifetime X 1 expires earlier than the lifetime X 2 is λ 1 /(λ 1 + λ 2 ).

Example B.1 A first-passage time problem

An electronic system has two crucial components, 1 and 2, that operate indepen-
dently of each other. The lifetime of component i has an exponential distribution
with mean 1/α i for i = 1, 2. If a component breaks down, it is replaced by a
new one. The time needed to replace component i by a new one is exponentially
distributed with mean 1/β i for i = 1, 2. The system continues to operate as long as
one of the components is functioning, but it fails when none of the two components
works. Both components are in perfect condition at time 0. What is the expected
time until the first system failure?
Let us say that the system is in state 1 (2) if only component 1 (2) is functioning
and it is in state 3 when both components are functioning. In view of the memory-
less property of the exponential distribution, we can define the random variable T i
as the time until the first system failure when the current state of the system is state
i. We wish to compute E(T 3 ). To do so, we derive a system of linear equations
in E(T i ) for i = 1, 2, 3. By conditioning on the next state and using the results in
(B.1), it follows that
1 β 2 1 β 1
E(T 1 ) = + E(T 3 ), E(T 2 ) = + E(T 3 ),
α 1 + β 2 α 1 + β 2 α 2 + β 1 α 2 + β 1
1 α 2 α 1
E(T 3 ) = + E(T 1 ) + E(T 2 ).
α 1 + α 2 α 1 + α 2 α 1 + α 2
These equations are easily solved for E(T 3 ).

The gamma distribution
A positive random variable X is said to be gamma (α, λ) distributed when it has
the probability density
α α−1
λ t −λt
f (t) = e , t ≥ 0,
Ŵ(α)
where α > 0 is the shape parameter and λ > 0 is the scale parameter. The symbol
Ŵ(α) denotes the complete gamma function which is defined by

∞

−t α−1
Ŵ(α) = e t dt, α > 0.
0
This function has the property that Ŵ(α + 1) = αŴ(α) for α > 0. In particular,
Ŵ(α) = (α − 1)! if α is a positive integer. The probability distribution function