Question

Let be a random variable with probability density function f(x) and moment-generating function 1 1 M(t) = =+ = ? 6 . 6 1 + - 1 36 + -e a) Calculate the mean = E(X) of X b) Calculate the variance o? = E(X -w' and the standard deviation of X

Answer 1

TUVA BRIL ic C Let y be the random variable with probability density function fans and the moment genereeting function is, met) = ţ + fet + 2++et. property in O E QAR) My los loco O E M 2 2 Malt) / 4.0 OE (22) de malts ico Now, E C#m.[t + fet + fettfety ()+(et) +54 (@t) + 5 m (et). = 075 et + $ét (+et (0) tet tjet + ext]. the 't3cmo+ 38340) TE teo + s + R + = 18333 0 to fund £42) - e. (my ct) to

LLLLL €62) - The C [ttet +5e24+5e?]] 0 = e c) +6340) Te +$**@Deo = + 9 +3 Ex3 = 4.5 To find out variance Var(x) = € ($27 - [ECX)]? = 4.5+ (1-8333)2 45-(3.3609) var(x) = 1-1391 and standard deviation of x is the square root of variance of x. sid 6) - J varc) = V1.1393 = 1.0673 =1.0673