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Philomena Haynes
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1 GNH7/GEOLGG9/GEOL2 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD TUTORIAL (6): EARTHQUAKE STATISTICS Question. Questions and Answers How many distinct 5-card hands can be dealt from a standard 52-card deck? Answer. Since the 5-card hand remains unchanged if you received the same 5 cards, but in a different order, the answer is, 52 C 5 = 52! / (7! 5!) = 2,598,96. Question 2. It is frequency assumed phenomena such as injury-causing accidents in a large industrial plant satisfy the three assumptions for a Poisson process. Suppose these assumptions are true for a particular plant, occurring at a rate of say, λ = ½ per week. What is the probability that eactly N = number of accidents will occur in the net 6 weeks? What is the mean number of accidents that should occur in this time and what is the variance? What is the probability of at least accidents? What is the probability of no accidents? Plot the histograms for µ= and µ=8. Answer 2 If N represents the number of such earthquakes to occur in the net t = 6 weeks, then N is a Poisson random variable with parameter µ = ½(6) = and its probability function is p N ( ) = e!, =,,2,... Both the mean and variance for N equal. The probability of eactly earthquakes in this period is p N () = e! =.22 And the probability of at least earthquakes is

/ (7! 5!) = 2,598,96. Question 2. It is frequency assumed phenomena such as injury-causing accidents in a large industrial plant satisfy the three assumptions for a Poisson process.

2 = e! =.5 The probability there will be no earthquakes is p N () = e! =.5 Sketch the histograms with µ= and µ=8. Mark on the mean and standard deviation...5. p() Question Plot the cumulative distribution function for the eample. Answer.75 F(t) t 2

3 Question Assuming that destructive earthquakes are occurring in a region at a rate of λ = ½ per decade, we will try and predict earthquakes occurrence from now, year 2, what is the epected number of years and the standard deviation for the first earthquake to occur? What is the probability that the first decade will be earthquake free? What is the probability that the first earthquake will occur in 25, and that it occurs in 28? Answer Given that destructive earthquakes are occurring in a region at a rate of λ = ½ per decade, this is equivalent to occurrences at the rate of /2 per year. We can predict the number of earthquakes we could epect to occur from now. If T is the number of years until the first earthquake occurs, then T is eponential with λ = /2. The epected number of years to the first earthquake is E[T] = (/2) - = 2 and the standard deviation for T is σ T = 2 years The probability that the first decade is earthquake free is: Pr[T>] = e - ()/2 =.67 The probability that the first earthquake occurs in 25 is Pr[ < T 5] = ( e - (5)/2 ) ( e - ()/2 ) =. and the probability that it occurs in 28 is Pr[7 < T 8] = ( e -(8)/2 ( e -(7)/2 ) =.5 Question 5 Show that the time interval distribution function, F T (), for slip-predictable behaviour in the Shimazaki model, with constant strain rate, k, is given by: F T () = ep[-(k/2). 2 ] Hint: Solution to the partial differential equation df()/d = - k.t.f() is f() = f().ep[(-k/2). 2 ].

What is the probability that the first earthquake will occur in 25, and that it occurs in 28?

4 Answer 5 We can t make assumptions about Poisson statistics and have to derive this from first principals. That is the whole point. In both the Shimazaki slip- and time-predictable models the strain rate is constant, k. For the slip-predictable model the probability of rupture is proportional to the accumulated strain, kt, where t is time. Given that no earthquake has occurred, the probability of an earthquake occurring in a small time interval t at time t is: We now consider time intervals, and to avoid confusion we change notation to. The probability that an earthquake occurs at any time,, we will denote P T (). The probability the interval T to the net earthquake will be greater than is the probability that the earthquake will occur in time + : Substituting: Pr[ N( t, t + t) = ] = kt t Pr[T > + ] = P T ( + ) = P T (). Pr[N(, + ) =] P T ( + ) = P T ().( k ) Now, So, P T ( + ) P T () = dp T (): dp T () = - k P T () d The solution of this partial differential equation is P T = P T () ep(-k 2 /2) or, since P T () =, F T () = ep(-ξ 2 ) where ξ = k/2. This is a Weibull distribution with eponent n = 2. So the Shimazaki model leads to a Weibull distribution. To get the probability density function we differentiate.

For the slip-predictable model the probability of rupture is proportional to the accumulated strain, kt, where t is time.

5 Supplementary Questions Question. Let us assume that the number of telephone calls to UCL are made in accordance with a Poisson process assumptions at the rate of 2 per hour during the period 9. to 2.. What is the epected number of calls during this period and the epected number in any single second? What is the probability of at least one call being made in any given minute and the probability of at least one call being made during any given second? Question 2. What is the probability of an earthquake occurring for the Shimazaki time-predictable model? What is the probability density function? 5

period 9. to 2.. What is the epected number of calls during this period and the epected number in any single second?

12.5: CHI-SQUARE GOODNESS OF FIT TESTS

12.5: CHI-SQUARE GOODNESS OF FIT TESTS 125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability