2 If the momentgenerating function of X is given by Mt exp5

2. If the moment-generating function of X is given by M(t) = exp{500t+5000t^2}, find the probability of P[27,060 (X 500)^2 50,240]. Make sure you provide all justification for your answer

Solution

M(t) = exp{500t+5000t^2}
  = exp [ 500t + 10000t^2 /2 ]......for normal distribution M(t) = exp [ mean*t + variance*t^2/2 ]...
so, this is a nornal distribution with mean = 500 and variance = 10000..i.e, s.d = 100..

P[27,060 (X 500)^2 50,240] = P [ 2.706 ( (X 500)^2 /10000 ) 5.024 ]
= P [ 2.706 Z^2 5.024 ] = P [  1.644992 Z   2.241428 ] = P [ Z   2.241428 ] - P [ Z 1.644992 ]
= 0.9875008-0.9500143 = 0.0374865 .....