Page 227 - Fundamentals of Probability and Statistics for Engineers

210 Fundamentals of Probability and Statistics for Engineers

f (y)
Y
1.5

σ 2 =0.1
X
1.0
σ 2 =0.3
X

σ 2
0.5 X =1.0

0.0 y
0 2 4 6 8

Figure 7.9 Lognormal distribution, f (y), with m X ˆ 0, for several values of 2 X

Y
quantities that are restricted to having only positive values. In addition, f (y)
Y
takes many different shapes for different values of m X 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 m X 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 f (y), we observe that, if medians of X and Y are
Y
denoted by X and Y , respectively, the definition of the median of a random

variable gives
0:5 ˆ P Y Y †ˆ P X ln Y †ˆ P X X †;

or
ln Y ˆ X : 7:45†

Since, owing to the symmetry of the normal distribution,

X ˆ m X ;

we can write

m X ˆ ln Y : 7:46†

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