Question

a) Let T be an exponentially distributed random variable with parameter l= 1. Let U be a uniformly distributed random variable. Use inversion to show how to calculate samples {tı, t2, ..} from samples {U1, U2, ..} of U. b) Use any software of your choice to estimate by Monte Carlo simulation: E[sin(tanh, LT))

Answer 1

- The pdf g exponential distribution with parameter dal is f(x) = e, xo to This implied cumulative distribution. Elmi) z let at distribution function The Cumulative distribution function of any eandom variable follows uniform distribution Hence we generate a uniform handom numbes first and then, equate L-ed with generated uniform number. The expression we get for a follows exponential distribution with Pasarmites 2.

se) u -ed =) loge" = log (1-4) = -x = log (1-u) =) * = -loge (1-4) This will be done in R.

#generating 1000 uniform numbers U=runif(1000,0,1) # inverting cdf to get exponential random numbers X=-log(1-U,base=exp(1)) # finding expectation of function given in question Z=sin(tanh(1/(1+x))) Mean=sum(Z)/1000 X#? print("expectation is ") print(Mean)

Only a snapshot of some observations and mean is enclosed.

Mean is.4917. Using the procedure of inversion we get n values for x corresponding to different n uniform random numbers

2:19 PM ... @ J1 4G or you will 4G y E (81 rextester.com/\/r_onlin: 3 : [936] 1.7628155250 0.6948714183 0.3748 896926 0.1843707889 1.3220226002 [941] 0.5893224235 0.1138649279 0.2425 846731 1.7742196369 0.3710850471 [946] 1.3568878243 0.2278549175 0.2530 434561 0.3747297744 0.1494458016 [951] 4.8946635109 1.4054440146 0.7591 856703 5.7360609217 0.1401664006 [956] 0.0182356717 0.5970655071 0.3538 767007 3.0933156172 0.2066208136 [961] 0.0501644265 0.2970230409 0.6663 397484 0.9975306130 0.4825661474 [966] 0.3054331198 1.0672905851 0.3554 658251 0.3194281071 0.5863958059 [971] 1.2949334947 0.0827757686 0.0608 191235 1.9538613468 0.9128463106 [976] 0.2282959407 0.3989073910 1.3122 558633 1.0043322690 0.1940007050 [981] 0.7457128971 0.4257320677 1.0001 829637 0.9315152414 1.4986025119 [986] 0.7644113950 0.4977486405 0.4571 313182 0.0565798209 0.2792843264 [991] 1.0653219131 0.2975864513 2.0531 455977 1.0854257720 2.0720429332 [996] 2.5882879483 0.9379984023 0.2316 978380 0.0923112619 0.0748992940 [1] "expectation is " [1] 0.4917684