3.5.
Hypoexponential Random Variables
Definition 1. A hypoexponential random variable is a sum of independent exponential
random variables. If 1, …, n are positive numbers then T1,…,n denotes the
hypoexponential random variable T1,…,n = T1 +  + Tn where T1, …, Tn are
independent and each Tj is an exponential random variable with decay rate j.
Proposition 1. Let m = (m1,...,mn) be a vector of positive integers and  = (1,...,n) be a
vector of positive numbers and let X1, …, Xn be independent gamma random variables
with Xj ~ (mj,1/j) and Y = X1 +  + Xn be their sum. Then Y is a hypoexponential
random variable.
Proof. This follows from the fact that each Xj is the sum of mj independent exponential
random variables each having decay rate j.
Since an exponential random variable is a gamma random variable, a hypoexponential
random variable is a sum of independent gamma random variables. Thus the density
function of a hypoexponential random variable is given by (4) in section 3.1 and also by
Moschopolos' formula (11) in section 3.3. However, it turns out that the density function
of a hypoexponential random variable can be expressed in terms of elementary functions
which is the purpose of this section. First we look at the simpler case where all the j in
Definition 1 are distinct. After this we look at the more complicated case where some of
the j are equal. The discussion in this case is based on the formulas of Mathai [7] and
Cetnar [9].
3.5 - 1
Proposition 1. Let 1, …, n be distinct positive numbers and T1,…,n = T1 +  + Tn
where T1, …, Tn are independent and each Tj is an exponential random variable with
decay rate j. Let An(t) = An(t; 1,…,n) be the density function of T1,…,n. Then


(1)
An(t; 1,2) =  1- 2 e-1t +  1- 2 e-2t
2
1
1
2
(2)
An(t;1,...,n) = c1e-1t +  + cne-nt
with
(3)
ck = ck(1,...,n) =
1n
(1-k)(k-1-k)(k+1-k)(n-k)
Proof. An(t; 1,2) = 1e-1t * 2e-2t. Doing the integration gives (1). Then (2) follows
from (1) and induction and the fact that convolution is associative and commutative. 
By Theorem 4 in section 3 the moment generating function of Y is
(4)
1m12m2  nmn
M(s) =
(1 - s)m1(2 - s)m2  (n - s)mn
Using partial fractions one can write
iik+1i,mi-1-k
M(s) =  
(i - s)k+1
i=1 k=0
n
(5)
mi-1
where
(6)

iik+1i,mi-1-k =  
s
(7)
imj
i = 
(j - i)mj
j=1
n
ji
So
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n
(8)
mi-1
A(t;m,) =  
i=1 k=0
iik+1i,mi-1-ktke-jt
k!
3.5 - 3