Titel_SS06

2

2





n@ fy

@

(12.15)

xis the sample mean of the observations, is the sample size assumed for the prior
distribution of

n'

R and n is the sample size for the new sample. In the present example

n@3.06.

Based on the new observations the posterior parameters are @@353.22and @@6.16. In

Figure 12.10 plots are shown for the prior and the posterior probability density functions for
y.

The likelihood of the observation can be established as:

5 2

1 2

11 ()ˆ

( ˆ) exp( )

2 2





 

y

y

yi f

fy

f

Lf



S

. @ @

(12.16)

The likelihood function is also shown in Figure 12.10. It is seen that the effect of the test
results is quite significant. The predictive probability density function for the steel yield stress
may according to e.g. Ditlevsen and Madsen (1996) be determined as:



2

ˆ^11

exp

2 2



y  

y

fyy

f

fff

! ! @@

" "

@@@ "$%$%@@@

#

2

(12.17)

where

^22
@@@  @@ fy (12.18)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

300

Mean value of steel yield stress

Probability desnity

320 340 360 380 400

0.08

0.09

0.10

Prior

Likelihood

Posterior

Figure 12.10: Illustration of prior and posterior probability density functions for the mean value of the
steel yield stress. Also the likelihood for the test results is shown.

In Figure 12.11 the predictive probability distribution and the probability distribution function
for the steel yield stress based on the prior information of the mean value are shown.