Download SOME IMPORTANT CONTINUOUS RANDOM VARIABLES

SOME IMPORTANT CONTINUOUS RANDOM
VARIABLES
Małgorzata Murat
UNIFORM DISTRIBUTION
Random variable X has uniform distribution U(a, b) if its
probability density function is given by



1
, x ∈ (a, b)
f (x) = b − a

0,
x ∈ R r (a, b)
Uniform distribution is very simple and used to model errors in
electrical communication with pulse code modulation.
1
expected mean: EX = (a + b)
2
1
variance: σ 2 = (b − a)2
12
EXPONENTIAL DISTRIBUTION
Random variable X has exponential distribution with parameter
λ > 0 if its probability density function is given by
f (x) =

λe−λx ,
0,
x­0
x<0
EXPONENTIAL DISTRIBUTION
The exponential distribution
is used for studies of reliability and of queuing theory,
which gives probability as a function of waiting time in a
queue for service
is related to the Poisson distribution
gives the probability distribution of the time between
successive random events for the same conditions as apply
to the Poisson distribution.
EXPONENTIAL DISTRIBUTION
expected mean: EX = λ
variance: σ 2 = λ2
NORMAL DISTRIBUTION
Random variable X has normal distribution with parameters µ,
σ > 0 if its probability density function is given by
f (x) =
σ
1
√
2π
(x − µ)2
−
2σ 2
e
expected mean: EX = µ determines the location of the
center of the distribution
variance: σ 2 determines the spread of the distribution