Fundamentals of Probability and Statistics for Engineers

quantities that are restricted to having only positive values. In addition, fY (y)
takes many different sh apes for different values of mX and X ( x > 0). As seen
from Figure 7.9, the pdf of Y is skewed to the right, this characteristic becoming
more pronounced as X increases.
It is noted that parameters mX and X appearing in the pdf of Y are the
mean and standard deviation of X, or ln Y, but not of Y. To obtain a more
natural pair of parameters for fY (y), we observe that, if medians of X and Y are
denoted by X and Y , respectively, the definition of the median of a random
variable gives

or

Since, owing to the symmetry of the normal distribution,

we can write

02 4 6 8

0.0

0.5

1.0

1.5

y

2

X= 0.1

2

X= 0.3

2

X= 1.0

fY(y)

Figure 7. 9 Lognormal distribution, fY (y), with mX 0, for several values of^2 X

210 Fundamentals of Probability and Statistics for Engineers

σ

σ

σ

ˆ 

 







0 : 5 ˆP…YY†ˆP…XlnY†ˆP…XX†;

lnYˆX: … 7 : 45 †

XˆmX;

mXˆlnY: … 7 : 46 †