Transcript exponential random variable

Chapter 4
Continuous Random Variables
& Probability Distribution
(Cont.)
Normal approximation to the
Binomial distribution
• Example 4-17:
• For many physical systems, the binomial
model is appropriate with an extremely large
values for n.
• In these cases, it is difficult to calculate
probabilities by using the binomial
distribution.
• Normal approximation is most effective in
these cases.
Normal approximation to the
Binomial distribution
• If X is a binomial random variable,
Z
X  np
np(1  p)
is approximately a standard normal random variable.
• The approximation is good for
np > 5 and n(1-p) > 5
• The approximation is good when n is large relative
to p.
• Examples: 4-18, & 4-19
Normal approximation to the
Poisson distributions
• Example: 4-20
• If X is a Poisson random variable with E(X) =  and
V(X) = ,
Z
X 

is approximately a standard normal random variable.
The approximation is good for  > 5.
Exponential distribution
• In Poisson distribution, X is the random variable
representing number of (flaws along a length of
wire).
• The distance between flaws is another random
variable of interest.
• The random variable X that equals the distance
between successive counts of a Poisson distribution
with means  > 0 is an exponential random
variable with parameter . The probability density
function of X is
f(x) =  e-x for 0  x < 
Probability distribution
function of exponential
random variable. Note
how the distribution is
skewed.
Exponential distribution (cont.)
• Cumulative distribution function:
F(x) = P(X  x) = 1 – e-x, x  0
• If a random variable X has an exponential
distribution with parameter ,
 = E(X) = 1/ 
2 = V(X) = 1/ 2
Example 4-21
Exponential distribution (cont.)
(1) Poisson process assumes that events occur uniformly
throughout the interval of observation, i.e., there is no cluster
of event. Thus, our starting point for observation does not
matter. This due to the fact that the number of events in an
interval of a Poisson process depends only on the length of the
interval, not on the location.
(2) In Poisson distribution, we assumed an interval could be
partitioned into small intervals that are independent. THUS,
knowledge of previous results does not affect the
probabilities of events in future subintervals. This is called
lack of memory property.
P(X<t1+t2|X>t1)=P(X<t2)
• The exponential distribution function is the only continuous
distribution with this property.
Example 4-22
P(X<t1+t2|X>t1)=P(X<t2)
Exponential distribution (cont.)
• Exponential distribution is often used in reliability studies as
the model for the time until failure of a device (taking into
consideration lack of memory property).
Example:
• Life time of a semiconductor chip might be modeled as an
exponential random variable with a mean of 40,000 hours.
Due to lack of memory property, regardless of how long the
device has been operating, the probability of a failure in the
next 1000 hours is the same as the probability of a failure in
the first 1000 hours of operation (i.e., the device does not
wear out).
Erlang distribution
• An exponential random variable describes the
length until the first count is obtained in a
Poisson distribution.
• Generalization of the exponential
distribution: The length until r counts occur in
a Poisson distribution. The random variable
that equals the interval length until r counts
occur in a Poisson process has an Erlang
random variable.
Erlang distribution (Cont.)
• The random variable X that equals the
interval length until r counts occur in a
Poisson process with mean  > 0 has an
Erlang random Variable with parameters  and
r . The probability density function of X is
f ( x) 
r 1  x
x e
r
(r  1)!
for x > 0 and r = 1, 2 , …
Example 4-24
Erlang distribution (Cont.)
• Erlang random variable with r = 1 is an
exponential random variable.
• An Erlang random variable can be represented
as the sum of r exponential random
variables.
• If X is an Erlang random variable with
parameters  and r,
 = E(X) = r/ 
2 = V(X) = r/ 2
Weibul distribution
• Weibul distribution is often used to model the time
until failure of many different physical systems.
• Distribution parameters provide a great deal of
flexibility to model systems in which the number of
failures:
– Increases with time (e.g., bearing wear).
– Decreases with time (some semiconductors).
– Remains constant (failures caused by external shocks to the
system).
Weibul distribution (Cont.)
• The random variable X with probability
density function
 x
f ( x)   
  
 1
  x  
exp    
    
for x > 0 is a Weibul random variable with
scale parameter  > 0 and shape parameter  >
0.
Weibul distribution (Cont.)
• By inspecting the probability density function,
it is seen that when  = 1, the Weibull
distribution is identical to the exponential
distribution.
• If X has a Weibull distribution with parameters
 and , then the cumulative distribution
function of X is:

F ( x)  1  e
x
 
 
Weibul distribution (Cont.)
• If X has a Weibul distribution with parameters  and
,

1
  E (x)  1  
 
2
2
1 
2
2 
2 
  V ( x)   1     1  
 
   
(r) = (r-1)!
Revision
• A Poisson random variable describes the number of
counts in an interval.
• An exponential random variable describes the
length until the first count is obtained in a Poisson
distribution.
• An Erlang random variable describes the length
until r counts occur in a Poisson distribution.
• Weibul distribution is used to model the time until
failure of many different physical systems.
ANNOUNCEMENTS
• Homework III:
3, 19, 27, 38, 46, 50, 61, 82
• Due on:
Monday, 16th of May