STAT213 Statistical Inference - Assignment 1

STAT213StatisticalInference-Assignment1 August17,2021 Question1 Foreachofthefollowing,determinewhetherthegivenfunctioncanserveastheprobabilitydistribution ofarandomvariablewiththegivenrange: a) f ( x )= x 2 5 for x =1 ; 2 ; 3 ; 4 ; 5 f (1)= 1 2 5 = 1 5 < 0 Notaprobabilitydistribution. b) f ( x )= x 2 30 for x =0 ; 1 ; 2 ; 3 ; 4 f ( x )= x 2 30 0 ; 8 x =0 ; 1 ; 2 ; 3 ; 4 4 X x =0 x 2 30 = 0+1+4+9+16 30 =1 f ( x ) canserveasaprobabilitydistribution. c) f ( x )= 1 5 for x =0 ; 1 ; 2 ; 3 ; 4 ; 5 f ( x )= 1 5 0 ; 8 x =0 ; 1 ; 2 ; 3 ; 4 ; 5 5 X x =0 1 5 = 1+1+1+1+1+1 5 = 6 5 > 1 Notaprobabilitydistribution. Question2 Theprobabilitydensityfunctionofarandomvariable X isgivenby f ( x )= ˆ kxe x 2 for x> 0 0 for x 0 Find a)thevalueof k Substitute u = x 2 and du =2 xdx 1= Z 1 0 kxe x 2 dx = k Z 1 0 1 2 e u du = k 2 e u 1 0 = k 2 k =2 b)thedistributionfunctionof X Z x 0 2 xe x 2 dx =2 Z u 0 1 2 e u du =1 e u =1 e x 2 F ( x )= ˆ 1 e x 2 for x> 0 0 for x 0 1 Question3 Considerthefollowingfunction f ( x )= x 2 +6 x 40 ; for x =1 ; 2 ; 3 ; 4 a)Verifythatitcanserveasthecumulativedistributionfunctionofadiscreterandomvariable F ( x )= 8 > > > > < > > > > : 0 for x< 1 7 40 for 1 x< 2 16 40 for 2 x< 3 27 40 for 3 x< 4 1 for x 4 b)˝ndtheprobabilitydistributionof X f ( x )= 8 > > < > > : 7 40 for x =1 9 40 for x =2 11 40 for x =3 13 40 for x =4 c)˝nd P (1 1 0 for x 1 checkwhetheritsmeanandvarianceexist. E ( X )= Z 1 1 2 x x 3 dx = Z 1 1 2 x 2 dx = 2 x 1 1 1 =2 Themeanexists. 0 2 = Z 1 1 2 x 2 x 3 dx = Z 1 1 2 x dx =2log x j 1 1 = 1 ˙ 2 doesnotexist. 2