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MATH 4030 – 6A
MORE DISTRIBUTIONS FOR
CONTINUOUS RV’S
1
• Uniform
• Log-Normal
• Gamma & Exp
• Beta
• Weibull
UNIFORM U(,) (SEC.
5.5):
f ( x)  C,   x   .
•
Value of C: Axiom of probability
•
Graphs of f(x) and F(x)
•
Probability
•
Mean E[X] and variance Var[X]
LOG-NORMAL DISTRIBUTION (SEC. 5.6):
Pdf:
ln x  
 1

1
2 2

, for x  0,   0,
f ( x)   2  x e

0,
otherwise .

2
3
Parameters:  and  > 0
LOG-NORMAL AND NORMAL:
When X has a Log-Normal distribution, then ln(X) has a
normal distribution with the same parameters.
X ~ LN  ,    ln X ~ N  ,  2 
Mean:

X  e
Variance:
 e
2
X
2   2
e
ln X   ,  ln2 X   2 .
2
2
2
.
ln( X )  ln X !

1 .
4
Probability:
P(a  X  b)  P(ln a  ln X  ln b)
GAMMA DISTRIBUTION GAMMA(,) (SEC. 5.7):
Parameters:  > 0 and  > 0
Pdf:
x

 1
 1

 
x
e
, for x  0,   0,   0,
f ( x)     

0,
otherwise .

where () is a value of the gamma function:

    x 1e  x dx
5
0
MEAN AND VARIANCE:
   .
   .
2
2

Used to model a
continuous random
variable that takes
positive values and has
asymmetric distribution.
(rainfall, energy
consumption, survival
time,…)
Represent a large family
of continuous
distributions with its
flexibility of two
parameters. (chi-square,
exponential, …)
6

EXPONENTIAL DISTRIBUTION:
A SPECIAL CASE OF GAMMA DISTRIBUTION
Exp(  )  Gamma(1,  )
Pdf:
 1  x
 e ,
f ( x)   
 0,

for x  0,   0,
otherwise .
Mean:
Variance:
  .
2
2
7
  .
EXPONENTIAL V.S. POISSON
continuous
discrete
Assuming 3 arrivals per hour on average.
• Let X be the number of arrivals in an hour,
then X has Poisson distribution with
parameter  = 3.
8
• Let Y be the time observed between two
arrivals. Then Y have exponential
distribution with parameter  = 1/3.
Beta Distribution Beta(,) (Sec. 5.8):
Pdf:
 x 1 1  x  1
     1
 1


x 1  x  , for 0  x  1,   0,   0,
f ( x )   B  ,  
  

0,
otherwise .

Parameters:  > 0 and  > 0


 
Variance:
.

 
.
2
        1
2
9
Mean:
10
Graph of Beta Distribution Beta(,):
The Weibull Distribution (Sec. 5.9)
Pdf:
x e
f ( x)  
0,

 1 x 
,
for x  0,   0,   0,
Parameters:  > 0 and  > 0.
otherwise .
x  1e x ,
f ( x)  
0,


1


Let y  x , x  y , dx 
for x  0,   0,   0,
otherwise .
1

1
y
CDF is F ( x)  1  e

1
x 
dy
, x  0.
Probability can be
easily calculated.
Mean and Variance:

   x   x
 1 x 
e
dx
(Let y  x  )
0


1 

u
0
1

e du  
u

1


1
1  .


2









2
1


2

   1    1   .
  
  




13
2
PROBABILITY DISTRIBUTIONS IN R
Distribution
Beta
Binomial
Chi-Square
Exponential
F
Gamma
Geometric
pbeta
pbinom
pchisq
pexp
pf
pgamma
pgeom
qbeta
qbinom
qchisq
qexp
qf
qgamma
qgeom
dbeta
dbinom
dchisq
dexp
df
dgamma
dgeom
rbeta
rbinom
rchisq
rexp
rf
rgamma
rgeom
Hypergeometric
phyper
qhyper
dhyper
rhyper
Log Normal
Negative
Binomial
Normal
Poisson
Student t
Uniform
Weibull
plnorm
qlnorm
dlnorm
rlnorm
pnbinom
qnbinom
dnbinom
rnbinom
pnorm
ppois
pt
punif
pweibull
qnorm
qpois
qt
qunif
qweibull
dnorm
dpois
dt
dunif
dweibull
rnorm
rpois
rt
runif
rweibull
Cumulative
probability
Functions
Quantile of
distribution
Probability value or
density value
Randomly generated
value from the distribution