topic 4 - continuous distributions basics of continuous distributions - pages 119 - 124 119 - 124...
TRANSCRIPT
Topic 4 - Continuous distributions
• Basics of continuous distributions - pages 119 - 124
• Uniform distribution - pages 135 – 136• Normal distribution - pages 125 - 131 • Gamma distribution - pages 138 - 141
Continuous Random Variables
• A continuous random variable can take on values from an entire interval of the real line.
• The probability density function (pdf) of a continuous random variable, X, is a function f(x) such that for a < b
• The cdf of X is defined as
( ) ( )b
a
P a X b f x dx
( ) ( ) ( )x
F x P X x f t dt
Some relationships
• What is the relationship between f and F?
• P(a ≤ X ≤ b) = F(b) – F(a)
• P(X = a) = P(a ≤ X ≤ a) = F(a) – F(a) = 0
Pipeline example• A pipeline is 100 miles long and every location
along the pipeline is equally likely to break• Let X be the distance measured in miles from
the pipeline origin where a break occurs• What is the cdf for X?
• What is the pdf for X?
• What is P(30 ≤ X ≤ 50)?
Requirements of a pdf
• A pdf must satisfy the following two requirements:
• Does the pipeline pdf satisfy these requirements?
( ) 0 for all
( ) 1
f x x
f x dx
Uniform distribution
• A uniform distribution on the interval from A to B, U(A,B), is defined by a pdf of the form
• Does f(x) meet requirements?
• What is the cdf for the Uniform distribution?
1( ) for f x A x B
B A
Mean and variance of a cont. random variable
2 2
2 2 2
( ( )) ( ) ( ) , expected value of ( )
( ), mean of or expected value of
[( ) ], variance of
( )
( ) ( ), moment generating function for
(0) ,
X
X X
X X
tXX
X X
E h X h x f x dx h X
E X X X
E X X
E X
M t E e X
M
2(0) ( )XM E X
Back to the Uniform
• What is the mean of a U(0,1) distribution?
• What is the variance of U(0,1) distribution?
Gamma distribution• The gamma distribution, is
defined by the following pdf
where
• Properties of the gamma function, – For – If is a positive integer, –
1 /1( ) , 0, 0, 0
( )xf x x e x
1
0
( ) for 0.xx e dx
Properties of the gamma distribution• Is it a valid pdf?
• Show
• Show =
( ) 1 (1 )XM t t
More on the gamma distribution• is called the shape parameter• is called the scale parameter• The exponential distribution is a special
case of the gamma with • The gamma distribution is used as a
probability model for the time or space before the th event in a Poisson process where events occur at the rate
• Gamma calculator
Back to the clunker car• Recall that my car breaks down once a week
on average. If the breakdowns occur as events in a Poisson process, then what is the probability less than a week passes before my first breakdown? Gamma or Poisson?
• Gamma Calculator
Pipe example• Defects along a piece of pipe occur as events
in a Poisson process with an average of 2 defects every 10 feet. What is the probability that the third defect will occur at least 20 feet from the beginning of the pipe?
• Gamma Calculator
Normal distribution
• The normal distribution, N(2), has a pdf given by
• The normal distribution is always bell shaped.
• The normal distribution is defined in terms of its mean and variance (standard deviation).
• Normal calculator
2
2
( )
21( ) -
2
x
f x e x
Weight gain example• The weight gain associated with an antidepressant is
normally distributed with a mean of 6 lbs and a standard deviation of 3 lbs.
• What is the probability of weight gain?• What is the probability of gaining between 0 and 12 lbs?• Normal Calculator
Standard normal distribution
• If X has a N(2) distribution, then Z=(X-)/ has a standard normal distribution, N(0,1).
• The standard normal is an important reference distribution.
• P(X ≤ x) = P(Z ≤ (x-)/) = (x-)/• The cdf of a standard normal, z), is tabled
in many textbooks• Standardized values, (x-)/indicate how
far in standard deviations the value x is from
• For any normal distribution, probabilities can be phrased in terms of standardized values
Empirical rule• What is the probability
– a normal falls within one standard deviation of the mean?
– a normal falls within two standard deviations of the mean?
– a normal falls within three standard deviations of the mean?
• Normal Calculator
Back to the weight gain example• Recall =6 and =3.• Using the empirical rule, answer the
following questions:– What is the probability of weight loss?
– What is the probability of a weight gain between 0 and 12 pounds?
Normal approximations
• Normal approximation to Binomial
• Normal approximation to Poisson
Do my data look normal?• In StatCrunch, a quantile-quantile plot (QQ
plot) plots ordered data values versus quantiles of a standard normal distribution.
• If the data are from a normal distribution, the points should lie approximately on a straight line.
• Concentration data
Other distributions
• The Weibull distribution and the log normal distribution are used to model failure times.
• The beta distribution is used to model proportions.
• There are many other distributions out there.
• Choose the one that serves as the best probability model for your setting.