Titel_SS06

Table 2.2. Assuming that the concrete cube compressive strength is Normal distributed it is
required to estimate the parameters on the basis of the experimental results.

It can be shown that the equations for the moments of a Normal distribution in terms of the
distribution parameters are given as:

1

2

2



 ^2

(

(

(2.96)

Analysing the sample data the first two sample moments are found as:

1

2

32.67

1083.36





m

m

(2.97)

The point estimates of the parameters , may now be determined by solving the equations:

22

32.67

1083.36









(2.98)

giving:

32.67

4.05









(2.99)

Using the method of maximum likelihood the maximum likelihood function is readily written
as:



2

2

2

1 1 1

11 ˆ-

()ˆ exp

2 2

n n

i

i

x

L

:

:  :

!!

"#"#"

$%"$%

x #

#

(2.100)

and correspondingly the log-likelihood function as



2

2

2

1 1 1

11 ˆ

() lnˆ

2 2

n i

i

x

ln

:

:  :

! 

"#"

$%

x  (2.101)

The mean values of the estimates may be determined by solving the following equations:





2

3 2

1111

2 2

211

(^1) ˆ
0
(^1) ˆ
-0





  













n i i n i i

ln

x

l

x

:

:::

:

::

(2.102)

yielding:



2 2 1 1 2 1

ˆ-

(^1) ˆ











n i i n i i

x

n

x

n

:

:

:

(2.103)

which by using the sample data gives: