Transcript Notes - De La Salle University

CONTINUOUS PROBABILITY
DISTRIBUTIONS
• Continuous Uniform Distribution
• Normal Distribution
• Gamma Distribution
- Erlang distribution
- Exponential distribution
• Chi-Square Distribution
• Lognormal Distribution
• Weibull Distribution
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Continuous Uniform Distribution
The density function of the continuous uniform random variable X on
the interval [A, B] is
1/(B-A),
AxB
F(x; A,B) =
0,
elsewhere
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Normal Distribution
• The most important continuous probability distribution in
the entire field of statistics
• Often referred to as the Gaussian distribution in honor of
Karl Friedrich Gauss (1777-1855) who derived its equation
from a study of errors in repeated measurements
Normal Distribution. The density function of the normal random variable X,
with mean  and variance 2, is
1
2
-(1/2)[(x
)/]
n(x; , ) = -------------- e
,
-  < x < ,
√2
where  = 3.14159… and e = 2.71828…
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Properties of the normal curve:
1. The mode, which is the point on the
horizontal axis where the curve is a
maximum, occurs at x = .
2. The curve is symmetric about a vertical
axis through the mean .
3. The curve has its points of inflection at x
=   , is concave downward if
 -  < X <  + , and is concave upward
otherwise.
4. The normal curve approaches the
horizontal axis asymptotically as we
proceed in either direction away from the
mean.
5. The total area under the curve and above
the horizontal axis is equal to 1.
ENGSTAT Notes of AM Fillone, De La Salle University-Manila

Figure 6.3 ???
• Two normal curves having the same
standard deviation but different means
• Identical in form but are centered at
different positions along the horizontal
Figure 6.4 ???
• Two normal curves with same means but
different standard deviations
• Centered at exactly the same position but
the curve with the larger standard deviation is
lower and spread out farther
Figure 6.5 ???
• Two normal curves having different means
and different standard deviations
• The probability associated with each
distribution are different
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
• The area under the curve bounded by the two coordinates x = x1 and x = x2
equals the probability that the random variable X assumes a value between x
= x1 and x = x2
• Difficulty encountered in solving integrals of normal density functions
necessitates the tabulation of normal curve areas
• Hopeless to attempt to set up separate tables for every conceivable values
of  and 
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
• The distribution of a normal random variable with mean 0 and variance 1 is
called a standard normal distribution.
• This can be done by means of the transformation
Z = (x - )/
ENGSTAT Notes of AM Fillone, De La Salle
University-Manila
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Boardwork: (17 student volunteers, 10 pts. each)
6.1/185) Given a standard normal distribution, find the area under the curve which lies
a) To the left of z = 1.43;
b) To the right of z = -0.89;
c) Between z = -2.16 and z = -0.65;
d) To the left of z = -1.39;
e) To the right of z = 1.96;
f) Between z = -0.48 and z = 1.74.
6.2/165) Find the value of z if the area under a standard normal curve
a) To the right of z is 0.3622;
b) To the left of z is 0.1131;
c) Between 0 and z, with z > 0, is 0.4838;
d) Between –z and z, with z > 0, is 0.9500.
6.7/186) A research scientist reports that mice will live an average of 40 months when their diets are
sharply restricted and then enriched with vitamins and proteins. Assuming that the lifetimes of such
mice are normally distributed with a standard deviation of 6.3 months, find the probability that a
given mouse will live
a) More than 32 months;
b) Less than 28 months;
c) Between 37 and 49 months.
6.10/186) The finished inside diameter of a piston ring is normally distributed with a mean of 10
centimeters and a standard deviation of 0.03 centimeters.
a) What proportion of rings will have inside diameters exceeding 10.075 centimeters?
b) What is the probability that a piston ring will have an inside diameter between 9.97 and 10.03
centimeters?
c) Below what value of inside diameter will 15% of the piston rings fall?
6.13/186) The average life of a certain type of small motor is 10 years with a standard deviation of 2
years. The manufacturer replaces free all motors that fail while under guarantee. If he is willing to
replace only 3% of the motors that fail, how long a guarantee should he offer? Assume that the
lifetime of a motor follows a normal distribution.
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Normal Approximation to the Binomial
If X is a binomial random variable with mean  = np and variance
2 = npq, then the limiting form of the distribution of
as n
, is the standard normal distribution n(z; 0,1).
Let X be a binomial random variable with parameters n and p. Then
x has approximately a normal distribution with  = np and 2 = npq
= np(1-p) and
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Example: Normal Approximation to the Binomial
A process for manufacturing an electronic component is 1% defective. A quality control
plan is to select 100 items from the process, and if non are defective, the process
continues. Use the normal approximation to the binomial to find
a) The probability that the process continues for the sampling plan described;
b) The probability that the process continues even if the process has gone bad (i.e. if the
frequency of defective components has shifted to 5.0% defective.
Solution:
Given:
a) p = .01, q = 0.99, n = 100, x = 0
a)
a) z = [0 + 0.5 – 100(.01)]/√100(.01)(.99) = -0.5025
a) P( z < 1) = 0.3085 (form Table A.3)
b)
b) z = [0 + 0.5 – 100(.05)]/√100(.05)(.95) = -2.065
b) P( z < 1) = 0.0197 (form Table A.3)
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Gamma Distribution
• The gamma function is defined by
• The continuous random variable X has a gamma distribution, with
parameters  and , it its density function is given by
• The mean and variance of the gamma distribution are
 = , and 2 = 2.
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Exponential Distribution
• The special gamma distribution for which  = 1 is called the
exponential distribution.
Exponential distribution curve
• The continuous random variable X has an exponential distribution, with
parameter , if its density function is given by
• The mean and variance of the exponential distribution are  = , and 2 = 2.
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Relationship of Exponential and Poisson Process
ENGSTAT Notes of AM Fillone, De La Salle
University-Manila
Gamma, Exponential and Erlang Distributions
• Applications – queuing theory, reliability problems, time
between arrivals at service facilities, and time to failure of
component parts and electrical systems
• In reliability theory, where equipment failure often conforms
to the Poisson Process,  is called mean time between failures.
• Many equipment breakdowns do follow the Poisson process,
and thus the exponential distribution does apply.
• Other applications include survival time in biomedical
experiments and computer response time.
• The binomial distribution also plays a role in exponential
and poisson distribution problems.
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Example: Gamma Distribution
Solution:
ENGSTAT Notes of AM Fillone, De La Salle
University-Manila
Examples: Exponential Distribution
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
ENGSTAT Notes of AM Fillone, De La Salle
University-Manila
Erlang Distribution
• A special case of the Gamma distribution where the shape
parameter  is an integer.
• As noted earlier, when parameter  equals 1, the distribution
is an exponential distribution.
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
ENGSTAT Notes of AM Fillone, De La Salle
University-Manila
Chi-Squared Distribution
• Also a special case of the gamma distribution which could be
obtained by letting  = /2 and  = 2, where  is a positive
integer.
where v is a positive integer.
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Lognormal Distribution
• The continuous random variable X has a lognormal
distribution if the random variable Y = ln (X) has a
normal distribution with mean  and standard
deviation .
• The resulting density function of X is
• The mean and variance of the lognormal distribution are
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Example: Lognormal Distribution
Rate data often follow a lognormal distribution. Average power usage (dB per hour) for a
particular company is studied and is known to have a lognormal distribution with
parameters  = 4 and  = 2. What is the probability that the company uses more than 270
dB during any particular hour?
Solution:
Given:  = 4 and  = 2
4
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
Weibull Distribution
• Introduced
by the Swedish physicist Waloddi Weibull in 1939.
• Used in reliability and life-testing problems such as the time
to failure or life length of a component, measured from some
specified time until it fails.
• Has the inherent flexibility that does not require the lack of
memory property of the exponential distribution.
ENGSTAT Notes of AM Fillone, De La Salle
University-Manila
Weibull Distribution
• The continuous random variable X
has a Weibull distribution, with
parameter  and  if its density
function is given by
where  > 0 and  > 0.
• The mean and variance of the Weibull distribution are
• The cumulative distribution function for the Weibull
distribution is given by
for  > 0 and  > 0
ENGSTAT Notes of AM Fillone, De La Salle University-Manila
ENGSTAT Notes of AM Fillone, De La Salle University-Manila