Page 123 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.37

[0, 1] then an inverse function exists, i.e., x = F −1 (u) for u ∈ [0, 1]. If U is a
X
uniform random variable in [0, 1] then find the distribution of the random
variable X = F −1 (u).
X
(b) Consider a transformation of random variables, X =− ln(U )/λ. Find the
output PDF, when U is a random variable uniformly distributed on [0, 1].
(c) Use the result in (a) and the rand function in Matlab to generate a real-
izations of an exponential random variable, i.e., ones that have a density
function

λ exp[−xλ] x ≥ 0
f X (x) = (3.59)
0 elsewhere
Hint: If U is uniform in the interval [0, 1] then 1 − U has the same distri-
bution. Generate 1000 realizations with λ = 1 and plot a histogram of the
resulting numbers.