On a Ratio of Functions of Exponential Random Variables and Some Applications

Consider L independent and identically distributed exponential random variables (r.vs) X
₁
, X
₂
,...,X
_L
and positive scalars b
₁
, b
₂
,...,b
_L
. In this letter, we present the probability density function (pdf), cumulative distribution function and the Laplace transform of the pdf of the composite r.v Z = (Σ
_(j=1)
^L
X
_j
)
²
/(Σ
_(j=1}
^L
b
_j
X
_j
). We show that the r.v Z appears in various communication systems such as i) maximal ratio combining of signals received over multiple channels with mismatched noise variances, ii) M-ary phase-shift keying with spatial diversity and imperfect channel estimation, and iii) coded multi-carrier code-division multiple access reception affected by an unknown narrow-band interference, and the statistics of the r.v Z derived here enable us to carry out the performance analysis of such systems in closed-form.