PROBLEM

Following are the times in days between successive earthquakes worldwide. An earthquake is included in the data set if its magnitude was at least 7.5 on a richter scale, or if over 1000 people were killed. Recording starts on 16th  of December 1902 and ends on 14th march 1997. There were 63 earthquakes recorded altogether and so 62 waiting times.

840,157,145,44,33,121,150,280,434,736,584,887,263,1901,695,294,562,721,76,

710,46,402,194,759,319,460,40,1336,335,1334,454,36,667,40,556,99,304,375,567,

139,780,203,436,30,384,129,9,209,599,83,832,328,246,1617,638,937,735,38,365,

92,82,220

The earth: structure, composition, and evaluation. Assume that the earthquake occurs at random and hence waiting times are exponentially distributed.

i)      Obtain the mean of the data set and estimate the parameter (λ).

ii)      Find empirical survival function and estimated survival function by replacing λ by its estimate.

iii)      Plot the empirical and the estimated exponential survival functions and comment on the adequacy of the exponential model to the data set.

Theory-

The exponential distribution is given by-

f(x) = λ {e}^{- λ x} , for x ≥ 0, where λ is the parameter of the distribution.

E[x]=1/ λ and hence λ=1/E[x].

Cumulative function (Distribution function) for f(x) is given by-

F(x)=  , hence for exponential distribution,

F(x) =  = 1-e-λx

And Survival Function is given by-

S(x)=1-F(x), hence

S(x)=1-(1-e-λx)= e-λx

R Code (Part-I)–

#Command to remove previous objects
rm(list=ls())

#given dataset
x=c(840,157,145,44,33,121,150,280,434,736,584,887,263,1901,695,294,562,721,76,710,46,
402,194,759,319,460,40,1336,335,1334,454,36,667,40,556,99,304,375,567,139,780,
203,436,30,384,129,9,209,599,83,832,328,246,1617,638,937,735,38,365,92,82,220
)

#Mean of the dataset
Mean=sum(x)/length(x)
cat("Mean time intervals between two earthquake is: ",Mean,".\n")

#calculating parameter Lambda
Lambda=1/Mean
Lambda

Output On R Console (Part)-

> #Command to remove previous objects
> rm(list=ls())
>
> #given dataset
> x=c(840,157,145,44,33,121,150,280,434,736,584,887,263,1901,695,294,562,721,76,710,46,
+ 402,194,759,319,460,40,1336,335,1334,454,36,667,40,556,99,304,375,567,139,780,
+ 203,436,30,384,129,9,209,599,83,832,328,246,1617,638,937,735,38,365,92,82,220
+ )
>
> #Mean of the dataset
> Mean=sum(x)/length(x)
> cat("Mean time intervals between two earthquake is: ",Mean,".\n")
Mean time intervals between two earthquake is:  436.8871 .
>
> #calculating parameter Lambda
> Lambda=1/Mean
> Lambda
[1] 0.002288921

Conclusion-

We can observe from above that mean of the given dataset is 436.8871 and hence estimated λ = 0.002288921.

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