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An overview of continuous distributions, including the uniform distribution, normal distribution, exponential distribution, and cauchy distribution. The probability density functions (pdf), cumulative distribution functions (cdf), moments, and properties of each distribution. Examples are given to illustrate the concepts. The normal distribution is further discussed, including the standard normal distribution and its importance in statistics.
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Continuous Distributions
Uniform or Rectangular Distribution: A rv X is said to follow a uniform
distribution over interval [a,b] if its pdf is of the form
f(x)=c if axb, and 0 elsewhere.
where c (>0) is a constant. We obtain c=1/(b-a)
So that we denote X~U[a,b].
Similarly we may define U(a,b), U(a,b], U[a,b).
2
1 1
r r
r
r
Normal Distribution: Of all theoretical continuous distributions, the
most important distribution is normal distribution.
Definition: A continuous rv X is said to follow a normal distribution with parameters
(,
2 ) (-, 0<<) if its density function is of the form:
We can easily verify that
f(x)>0 and
x
x
f x ,
exp
2
We denote X~N(,
2 )
Moments: E(X)=
2 =
2
Result: For r=1,2,…
2r
=E(X- )
2r =(2r-1)(2r-3)…5.3.1
2r
2r +
=E(X- )
Result: Mode of X is .
Note: Mode is that value of x for which f(x) is maximum.
Result: If X~N( ,
2 ), then Z=(X- )/ ~N(0,1).
Z is called the standard normal rv and distribution of Z is called
standard normal distribution.
The pdf of standard normal distribution is given by
We denote the cdf of Z by
Normal probability tables for (z) are available.) are available.
2
2
1
z
2
2
1
z
y
z e dy
x
2
2
1
What is the probability that the signal is larger than 240 micro volts
given that it is larger than 210 micro volts? If 20 such signals are sent,
What is the probability that not more than 2 will exceed 240 micro
volts?
Ex: A machine produces bolts in a length (in cm) found to obey a
normal distribution N(10,.01). The specifications for bolt call for an
item within a length 10.050.12. A bolt not meeting these
specifications is called defective. What is the probability that out of 20
bolts produced by the machine, none is defective?
9.5 9.7 9.9 10.1 10.3 10.
X
0
1
2
3
4
f(x)
Exponential Distribution: Exponential distribution plays an
important role in reliability analysis and queueing theory.
Definition: A continuous rv X is said to follow an exponential
The cdf of exponential distribution is given by
Mean: E(X)=
r
th moment about origin: r
r .r!
Variance: Var(X)=
2
otherwise
e if x
f x
x
0
, 0 , 0
1
( )
/
/
x
Result: If life time of a component follows an exponential distribution,
then R x
(t)=R(t), i.e., P(X>t+x|X>x)=P(X>t).
Note: If X is the life time of a component with parameter , then
is the mean life of the component.
Cauchy Distribution: A rv X is said to follow a Cauchy distribution if
its pdf is of the form
Moments: For Cauchy distribution and r
’ does not exist for r=1, 2,3,….
For instance
Which is not convergent.
2
dy
y
dx
x
x
E X
1
( 1 )
( ) 2
Ex: For a non-negative function g(X) of X for which E[g(X)] exists and h>
(i) P[g(X)
