Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 8

300

n = 1000
Normal PDF
with μ = 100 and σ = 20

250

200

150

100

50

0 0

50 100

y

150 200

Figure 1.3 Histogram of Y. The solid line is the normal PDF with μ = 100 kPa and σ = 20 kPa.

1.2.3 provides a more robust method of assessing goodness of fit to a normal distribution,
although it is less visually appealing.

1.2.2.2 Cumulative distribution function
The cumulative distribution function (CDF) of Y is defined as the probability that a ran-
dom variable Y is less than or equal to a specific numerical value y . In this chapter, an
uppercase symbol denotes a random variable and a lowercase symbol denotes an ordinary
variable (it is helpful to imagine this ordinary variable taking a specific constant number,
say 100 kPa). This distinction is critical. Symbolically, the CDF is denoted by F( y ). Basically,
F( y ) is the integral of the PDF (or PDF is the derivative of CDF). For the normal distribution,
its CDF is

y

1 −−

⋅

( t µ

σ

)

2

⋅

∫

(1.5)

F () y = exp dt

2

2

2 πσ ⋅

−∞

(a) 700 (b) 40

35

600

30

500

25

400

20

300

15

200

10

100

5

0 0

0 50 100

y

150 200 0 50 100

y

150 200

Five bins 100 bins

Figure 1.4 Histograms of Y with two different number of bins.