Question

5.7. The density function of X is given by Man = fat we Sa+ bx2 0 < x < 1 otherwise If E[X] = 3, find a and b.

Answer 1

The density function of x f(x)= Sa+bx² lot is given by okacit otherwise I f(x) dx = 1 itbx 2) dx + tt f o da a R1(0,1) ( ax + bx®)! = = 0+b= =) 3 a+b=3 - 0 E[XJ = (x fox) da Excatba 2 dat Ixco) da RICO,1) 9 2 2 + b = 4 2a+b 7

.. 2075 = 3 = 10a+56=12 © x5- gives 5a = 3 = 10 = 315 b= 3(1-9) = 3(1-3) = 8 16=6/5) Now, 1X-0.6l> 0. I 1 X-0.620 X-0.6 = 1X-0.61 0.1 = x >0.7. X-0.620 > - X-0.6) = 1X-0.6L>0.) = x <0.5 .. P (1X-0.6170,1)= P(X70.7) + P (X0.5) = f(x) dx + f f(a) da 0.7 I cat bx dx + locat bar 2 da 00 0.5 0.7

0.343b + 0.5 at 0.125 b at b -0.7a 3 3 3 3. P (IX-0.610.1) = (ox+bge)+ ( oz+by?) = 0+1 -0.74 - 0.3436 + 0.5a + 0.1256 = (1-0.7+0.5 )a + (1-0.343 +0.)25) b. - - 0.8 (3) + 0.782 19.) = 0.7928 3 51 IP (1x-0.6170.))=0.7928

F(x) = 1 f(x) dx If aco, f(a)=0 a if os x 21, Fy(x) = Fx(x) = 0 eat bx²) dax- = axtbas = 3x + 6 x² 5 5 3

Fy@) = 4(3x+20) If x21 Fx (x) = f'(a+bx²) dx = (ax + bx3 )' = atb L = + = 1 Thus, calf of X is given by Coif. x so - Fx (x) = { 1 (3x+228) if osasi it all