Continuous Random variable

Continuous Random VariablesContinuous Random Variables
Prepared By :
Patel Jay C
ME(EC)-140870705004

DefinitionsDefinitions
• Distribution function:
• If FX(x) is a continuous function of x, then X is a
continuous random variable.
o FX(x): discrete in x  Discrete rv’s
o FX(x): piecewise continuous  Mixed rv’s
•

Equivalence:
• CDF (cumulative distribution function)
• PDF (probability distribution function)
• Distribution function
• FX(x) or FX(t) or F(t)

Probability Density Function (pdf)Probability Density Function (pdf)
• X : continuous rv, then,
• pdf properties:
1.
2.

• Equivalence: pdf
o probability density function
o density function
o density
o f(t) =
dt
dF
,)(
)()(
0∫
∫
=
=
∞−
t
t
dxxf
dxxftF
For a non-negative
random variable

Exponential DistributionExponential Distribution
• Arises commonly in reliability & queuing theory.
• A non-negative random variable
• It exhibits memoryless (Markov) property.
• Related to (the discrete) Poisson distribution
o Interarrival time between two IP packets (or voice calls)
o Time to failure, time to repair etc.
• Mathematically (CDF and pdf, respectively):

CDF of exponentially distributedCDF of exponentially distributed
random variable withrandom variable with λλ = 0.0001= 0.0001
t
F(t)
12500 25000 37500 50000

Exponential Density Function (pdf)Exponential Density Function (pdf)
f(t)
t

Memoryless propertyMemoryless property
• Assume X > t. We have observed that the component
has not failed until time t.
• Let Y = X - t , the remaining (residual) lifetime
• The distribution of the remaining life, Y, does not
depend on how long the component has been operating.
Distribution of Y is identical to that of X.

• Assume X > t. We have observed that the component has
not failed until time t.
• Let Y = X - t , the remaining (residual) lifetime
y
t
e
tXP
tyXtP
tXtyXP
tXyYPyG
λ−
−=
>
+≤<
=
>+≤=
>≤=
1
)(
)(
)|(
)|()(

• Thus Gt(y) is independent of t and is identical
to the original exponential distribution of X.
• The distribution of the remaining life does
not depend on how long the component has
been operating.
• Its eventual breakdown is the result of some
suddenly appearing failure, not of gradual
deterioration.

Uniform Random VariableUniform Random Variable
• All (pseudo) random generators generate
random deviates of U(0,1) distribution; that
is, if you generate a large number of
random variables and plot their empirical
distribution function, it will approach this
distribution in the limit.
• U(a,b)  pdf constant over the (a,b) interval
and CDF is the ramp function

Uniform densityUniform density
U(0,1) pdf
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
time
cdf

Uniform distributionUniform distribution
• The distribution function is given by:
{
0 , x < a,
F(x)= , a < x < b
1 , x > b.ab
ax
−
−

Uniform distributionUniform distribution
U(0,1) cdf
0
0.2
0.4
0.6
0.8
1
1.2
0
0.08
0.16
0.24
0.32
0.4
0.48
0.56
0.64
0.72
0.8
0.88
0.96
1
1.04
1.08
1.16
1.24
1.32
1.4
1.48
time
cdf
U(

a b
F(x)
x0
1
F(x) = 0, for x < a
F(b) = 1, for x > b
It is also possible to use the cumulative probability function to calculate probability.
The probability is 0 for any value under a, and 1 for any value over b.
F(x) = P(X ≤ x) for all x.
Cumulative Distibution Function F(x)Cumulative Distibution Function F(x)

a b
F(x)
x
To find P(x ≤ c)
c
c da b
F(x)
x
To find P(c ≤ x ≤ d)
P(x ≤ c) = F(c)
P(c ≤ x ≤ d) = F(d) – F(c)

Continuous Random variable

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