Let X be the strength of a certain material where X ey and

Let X be the strength of a certain material, where X = e^(y) and Y is normally distributed with mean 10 and variance 1. Evaluate the probability that 10,000 < X < 20,000.

Solution

X = e^(y), so log x =y

10,000 < X < 20,000, taking log this becomes, log 10,000 < log X < log 20,000

=9.2103 < logx < 9.9034

Z value for 9.2103, z=(9.2103-10)/1 =-0.79

Z value for 9.9034, z=(9.9034-10)/1 =-0.10

P(10,000 < X < 20,000)

=P( -0.79 <z <-0.10)

=P( -0.10) – P(-0.79)

=0.4602-0.2148

=0.2454

Alternate method

If we calculate directly from lognormal distribution from software,

P(10,000 < X < 20,000)

P( x < 20000) - P(x < 10000)

=0.4616- 0.2149

=0.2467