251greatD 3/31/06
Great Distributions I Have Known
Note that F 5  Px  5 . The following distributions are all that are usually used in first-year Statistics.
Distribution
Uses
Formula
Mean
Variance
Discrete
Distributions
Binomial
Gives
Px  C xn p x q n x
  np
probability of x
 2  npq
No formula for
successes in n
F x  .
tries. p is
Never use Px 
constant
formula if a table
probability of
success on 1 try. is applicable.
Use to replace
h-g when
N  20 n .
Binomial
Gives
See above. Since
probability of
pq
E p   p
x
Var p  
p  or x  pn
p as
n
n
proportion of
successes in n
tries
Geometric
Gives
q
1
P( x)  q x1 p
2  2

probability that
p
p
F x 1  q x
the first success
occurs on try x .
Poisson
Gives
e m m x
m


P
x

probability of x
2 m
x!
successes in an
m is called the
interval in
parameter of the
which the
Poisson
average number
distribution.
of successes is
No formula for
m.
F x  .
Use to replace
Never use Px 
Binomial with
m  np when
formula if a table
is applicable.
n
 500 .
p
Hypergeometric
Gives the
probability of x
successes in a
sample of n
taken from a
population of
N in which
there are M
successes.
Px  
C nNxM C xM
C nN
No formula for
F x  .
To get F x  add
P x  .
If p 
  np
M
,
N
2 
N n
npq
N 1
Great Distributions I Have Known (pg. 2)
1
Distribution
Continuous
Distributions
Continuous
Uniform
Exponential
Uses
Gives the
probability that
x
Lies on an
interval when
you are told that
the uniform
distribution
between c and
d applies.
x is usually the
amount of time
you have to wait
until a success.
Formula
f x  
1
d c
for c  x  d ,
Mean

Variance
cd
2
2 
1
c

d  c 2
f x   0
otherwise.
xc
F x  
d c
for c  x  d ,
12
F x   0 for x  c
and
F x   1 for x  d
f x   ce cx and
F x  1  ecx
when x  0 and
the mean time to a
1
. Both
c
are zero if x  0 .
1
 1 z2
f z  
e 2
2
x
.
z

1
c
success is
Standardized
Normal
This is the
distribution of
x when  is
known.
It is used to
approximate the
binomial dist.
when np  5
and nq  5 with
  np and
0
1

Always use tables
once you have
values of z .
  npq .
It is used to
approximate the
Poisson
Distribution
when m  25
with   m and
  m.
t distribution
This is the
distribution of
x when  is
unknown but s
is known.
Use tables.
0
DF
DF  2
2
Great Distributions I Have Known (Page 3)
Distribution
Chi-squared
 2 distribution
 
F distribution
Uses
Distribution of
s 2 when x is
normal and 
is known.
The ratio of
2
two  ' s from
the same
population has
the F
distribution. It is
used to compare
variances.
Formula
Mean
Variance
Use tables
mean  DF
var iance  2DF
Use tables
DF2
DF2  1
2DF2 2 DF1  DF2  2
DF1 DF2  22 DF2  4
Remember:
(i) If you are looking for numbers of successes when the number of tries is given
and the probability of success is constant, you want the Binomial
distribution. This distribution can also be used to replace the Hypergeometric
when the population size is large N  20 n  .
(ii) If you are looking for the try on which the first success occurs out of many
possible tries when the probability of success is constant, you want
the Geometric distribution.
(iii) If you are looking for numbers of successes when the average number of
successes per unit time or space is given, you want the Poisson
distribution. This distribution can be used to replace the binomial when the
n

population is large and the probability of success is small   500  .
p

(iv) If you are looking for numbers of successes when the number of tries is given
and the probability of success is not constant because the total number
of successes in the population is limited, you want the Hypergeometric
distribution.
(v) If you are interested in the distribution of x and know  , use the Normal
distribution with a standard deviation of  x ; if you are interested in the
distribution of x and know s , use the t distribution with a standard deviation
of s x . This also works for differences between means.
(vi) If you are interested in the distribution of the sample variance and the parent
distribution is approximately Normal, use the Chi-squared  2 distribution;
If you are comparing two variances, use the F distribution.
(vii) The following substitutions are commonly used for distributions. You must show
that the conditions for use (under ‘if’) are satisfied. See next page for
table.
 
3
Great Distributions I Have Known (Page 4)
Replace
Binomial
With
Poisson with m  np
Hypergeometric
Binomial with p 
Binomial
M
N
Normal with   np ,
If
n
 500
p
N  20 n
np  5 and nq  5
  npq
Poisson
Normal with   m ,
if m  25
 m
Hypergeometric
Normal p 
,   np ,

M
N
np  5 and nq  5 , but
think about Binomial if
N  20 n
N n
npq
N 1
4