X = rexp(sims, lambda) #define t (in an interval near 0) t = seq( from = 0, to = 2, length.out = 10) #calculate the empirical and analytical MGFs MGF.a = sapply(t, function(t) lambda /(lambda - t)) MGF.e = sapply(t, function(t) mean( exp(t *X))) #plot the MGFs, should match plot(t, MGF.e, main = 'Exponential MGF', xlab = 't', ylab = '', type = 'l', col = 'black', lwd = 3) lines(t, MGF. #replicate set.seed( 110) sims = 1000 #define a simple parameter lambda = 5 #generate the r.v. 9 Limit Theorems and Conditional Expectationīefore we dive head first into MGFs, we should formally define statistical ‘moments’ and even freshen up on our Taylor Series, as these will prove especially handy when dealing with MGFs (and have likely already proven handy in the PMF problems of previous chapters).

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Demo sims 3 generation