Transcript Normal distribution

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
b
P (a  X  b ) 
 f (x )dx
a
• The cdf of X is defined as
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:
f (x )  0 for all x

 f (x )dx  1

• Does the pipeline pdf satisfy these requirements?
Uniform distribution
• A uniform distribution on the interval from
A to B, U(A,B), is defined by a pdf of the
form
1
f (x ) 
B A
for A  x  B
• Does f(x) meet requirements?
• What is the cdf for the Uniform distribution?
Mean and variance of a cont. random variable

 E (h (X ))   h (x ) f (x )dx , expected value of h ( X )

 X  E (X ), mean of X or expected value of X
   E [(X  X ) ], variance of X
2
X
2
   E (X )  X
2
X
2
2
 M X (t )  E (e ), moment generating function for X
tX
 M X (0)  X , M X (0)  E (X )
2
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, G(a,b), is defined
by the following pdf
1
f (x )  a
x a 1e  x / b , x  0,a > 0, b > 0
b G(a )

where G(a )   x a 1e  x dx for a > 0.
0
• Properties of the gamma function, G(a)
– For a > 1, G(a)  (a1)G(a1)
– If a is a positive integer, G(a)  (a1)!
– G(1/2)  
Properties of the gamma distribution
• Is it a valid pdf?
• Show M X (t )  1 (1  b t )a
• Show  = ab
More on the gamma distribution
• a is called the shape parameter
• b is called the scale parameter
• The exponential distribution is a special
case of the gamma with a 1.
• The gamma distribution is used as a
probability model for the time or space
before the ath event in a Poisson process
where events occur at the rate b1/l.
• 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
f (x ) 
1
e
2
( x   )2
2 2
-  x  
• The normal distribution is always bell
shaped.
• The normal distribution is defined in terms
of its mean and variance (standard
deviation).
• Normal calculator
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-)/) = F((x-)/)
• The cdf of a standard normal, F(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.