Download Distributions

Distributions
Probability Distributions
BUS-121
November, 2013
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Probability Distributions
This short PPT is intended to help students with their
understanding and learning of probability distributions used
in this course BUS-121.
This presentation cannot be used for understanding without
reference to the full presentations, the course textbooks or
tutors.
It is intended as a useful summary of the main
characteristics of each selected distribution but without
explanation or attempt at full description.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Probability Distributions
A probability distribution is a table, formula, or graph that
describes the values of a random variable and the probability
associated with these values.
Since we’re describing a random variable (which can be
discrete or continuous) we have two types of probability
distributions:
– Discrete Probability Distribution, (Chapter 7) and
– Continuous Probability Distribution (Chapter 8)
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Distributions of Each Type
Discrete
Continuous
Binomial
Normal
Poisson
Uniform
Exponential
Students’ t
Chi Squared
F
These are the distributions to learn for this course
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Binomial Distribution
Statisticians have developed general formulas for the mean,
variance, and standard deviation of a binomial random
variable.
The Binomial Distribution can be approximated by the Normal
Distribution for large values of n
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Poisson Distribution
The probability that a Poisson random variable assumes a
value of x is given by:
and e is the natural logarithm base.
The mean and variance are the same value
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
The Normal Distribution
Important things to note:
The normal distribution is fully defined by two parameters:
its standard deviation and mean
The normal distribution is bell shaped and
symmetrical about the mean
Unlike the range of the uniform distribution (a ≤ x ≤ b)
Normal distributions range from minus infinity to plus infinity
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Standard Normal Distribution
A normal distribution whose mean is zero and standard
deviation is one is called the standard normal distribution.
0
1
1
As we shall see shortly, any normal distribution can be
converted to a standard normal distribution with simple
algebra. This makes calculations much easier.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Uniform Distribution
Consider the uniform probability distribution (sometimes
called the rectangular probability distribution).
It is described by the function:
f(x)
a
b
x
area = width x height = (b – a) x
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
=1
Exponential Distribution
Another important continuous distribution is the exponential
distribution which has this probability density function:
Note that x ≥ 0. Time (for example) is a non-negative quantity; the
exponential distribution is often used for time related phenomena such
as the length of time between phone calls or between parts arriving at
an assembly station.
For the exponential random variable:
The mean and standard deviation are the same value
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Student t Distribution…
Here the letter t is used to represent the random variable,
hence the name. The density function for the Student t
distribution is as follows…
ν (nu) is called the degrees of freedom, and
Γ (Gamma function) is Γ(k)=(k-1)(k-2)…(2)(1)
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
8.11
Student t Distribution…
Much like the standard normal distribution, the Student t
distribution is “mound” shaped and symmetrical about its
mean of zero:
The mean and variance of a Student t random variable are
E(t) = 0
and
V(t) =
for ν > 2.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chi-Squared Distribution…
The chi-squared density function is given by:
As before, the parameter
freedom.
is the number of degrees of
Figure 8.27
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
F Distribution…
The F density function is given by:
F > 0. Two parameters define this distribution, and like
we’ve already seen these are again degrees of freedom.
is the “numerator” degrees of freedom and
is the “denominator” degrees of freedom.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
F Distribution…
The mean and variance of an F random variable are given
by:
and
The F distribution is similar to the distribution in that its
starts at zero (is non-negative) and is not symmetrical.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Central Limit Theorem (Chapter 9)
The sampling distribution of the mean of a random sample
drawn from any population is approximately normal for a
sufficiently large sample size.
The larger the sample size, the more closely the sampling
distribution of X will resemble a normal distribution.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Central Limit Theorem (Chapter 9)
If the population is normal, then X is normally distributed
for all values of n.
If the population is non-normal, then X is approximately
normal only for larger values of n.
In most practical situations, a sample size of 30 may be
sufficiently large to allow us to use the normal distribution as
an approximation for the sampling distribution of X.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.