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Topic 3-6 Log-normal distribution

Solved problems

Problem 3-6-1

The time between severe earthquakes at a given region follows a log-normal distribution with a coefficient of

variation of !"# The e$pected time between severe earthquakes is %! &ears#

'a( )etermine the parameters of this log-normall& distributed interval time T# 'ans# #3!%* !#3%+(

'b( )etermine the probabilit& that a severe earthquake will occur within ,! &ears from the previous one#

'ans# !#!!!33(

'c( Suppose the last severe earthquake in the region took place 1!! &ears ago# hat is the probabilit& that asevere earthquake will occur over the ne$t &ear. 'ans# !#!3(

Solution/

'a( The 0parameters0and of a log-normal #2# are related to its mean and standard deviationand as follows/

,

,

,

1

,

= +

=

ln' (

ln

Substituting the given values

T

T

T

T = =! %!# * * we find

, !#1%, and #3!4%14* hence

0.385 and4.308

The importance of these parameters is that the& are the standard deviation and mean of the related variable

Xln'T(* whileXis normal# Probabilit& calculations concerning Tcan be done throughX* as follows/

'b( P'T,!( P'ln Tln ,!( P'X- ln ,!(

P'X

ln,!(

P'Z-3#!+441434(

0.00033

'c( 0T5 1!!0 is the given event* while 0a severe quake occurs over the ne$t &ear0 is the event 01!! T 1!10*

hence we seek the conditional probabilit&

P'1!! T 1!1 7 T5 1!!(

P'1!! T 1!1 and T5 1!!( 8 P'T5 1!!( P'1!! T 1!1( 8 P'T5 1!!(

P'ln 1!! ln T ln 1!1( 8 P'ln T5 ln 1!!(

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P'ln ln1!! 1!1

<

<

X( 8 P'

X>

ln1!!(

9P' Z to 1+"* or

'DD( increase the ordinar& probabilit& of pressure at F within the normal range to ;+"#

hich option is better in terms of minimiGing the probabilit& of unsatisfactor& water service to the cit&.

'ans# D(