Question

Question Let X be a continuous random variable with the following probability density function (pdf) 0.5e fx (x) = { 0.5e-1 x < 0. <>0.. (a) Show that fx (x) is a valid pdf. (b) Find the cumulative distribution function Fx (.x). (e) Find F='(X). (d) Write an algorithm to generate a sample of size 1000 from the distribution of X using the inverse-transform method. Be as precise as possible.

Answer 1

Question. Let x be a continuous random Variable with the following probability density function (pdf) a f (x) = 50.5e acor d - 0.52,0 a) show that fx (x) is a valid Pdfor To show force) to be a valid pdf, we have to show that fx (re) sadistys the following property I f (G) 7,0 VREIR D oog die = 1 boot consider, og To for Che) - Soiset 0.5 2 250 227,0 . disa We know that e 70 V 20EIR. 0.5elo & 05 elo VrEIR . fx (a) ZO VYFIR . consider, fx (e) dx = 10. 5 é dae + Tose are 10. setT+0.5 el LIT-o [-1 To

- 0.5(1-0) - 05(0-1) br - (since é o = 0.5 +0.5= 1. (fx Cae) dae = 1 . foca) is a valid pidif find the cumulative distribution function Ex(x). Fx(x) = (fore) de . case PESO if x 70 h aline To. see dut 70.5 e da + 0.5 - Jo - 05/1-OJ + 0. 5L-e 7 = 0.5 + 0.5-0.5e = 1-0. 5e

case it if ecool er fx(x) = ( fr (4) de (0.5e" de = 0.5[ete o.se r(e) = r 0.5et eco 1-0. 5e 27/0 Find Fr() a Here x is continuous random Variable & we know fx che) is non-decreasing (Here strictly increasing due to continuous nature of x). This implies inverse of Ę exist. For eco For a70 2 let Ta T o ise = 4 clont To.se - 4 h e = 24 h 0. 52 = 1-4 ne = log24 " = 2(1-4) above Relk and beloce & Corral) x = log to E (2) Tog2re S o Note: The has different meaning log oft x 7 % 2 (1 )

Write an algorithm to generate size 1000 from the distribution the inverse transfoom method. we have a sample of X of using to f log 2 x < 6 log qi-Rey 27 / Algorithm: Draw a random of size 10oo. Or table to geneoate Let this sample sample from Uniform (0, 1). use random number generator 1000 samples from uro, 1) be 4, 42,..., 44000 use Xi = log24; and use x;= log i n 2/1-4;) to generate new sample xi, i=1,2,.., 1000 from desire given distribution. Forex if (i = 0.78 then Xi = log/ 1 21-0.78) - log 0.44 = -log 0.44 = -(-0.821)=0.821 if 4i = 0.09, Xie 10g[2.(0.09)= -1.71 tepead this procedure 1000 times to get desire 3) sample