The table below shows some data on the times (in days) between serious earthquakes. An earthquake is included if its magnitude is at least 7.5 on theRichter scale or if over 1000 people were killed. Recording starts on 16 December 1902 (4500 killed in Turkistan). The table includes data on 21earthquakes, that is, 20 "waiting times" between earthquakes.

Assume X is the time between serious earthquakes (in days) and that X -Exp(1). Use your data to estimate à. The transformation Y = VX is applied to reduce the spread of the data. On paper, derive the probability density function for Y, and evaluate this at the values given in the table below, using your estimated value of å from part (a) (give your answers correct to 3 decimal places):

Use R to simulate 10° realisations from Y. Use this sample to estimate Pr(Y > 6.2). Use R to simulate 10° realisations from Y. Use this sample to estimate Pr(Y > 6.2).