Suppose that X has pdf Find EX Find VarianceX Find E exp 12

Suppose that X has pdf Find E(X) Find Variance(X) Find E (exp [1/2 X]), i.e., find E (e^0.5 X).

Solution

E(x) = Integral of x.f(x) from negative infinity to zero

So,

E(x) = Integral of (x.e^x dx) = (x-1)e^x from -infinity to zero

which is -1

Variance is given by:

Integral of (x² f(x) ) - (mean)²

= Integral of x² e^xdx - (-1)²

= [x² (e^x ) - 2(x-1)e^x from -infinity to 0 ] - (-1)²

= 2 - 1

= 1

c)

E ( e^0.5X ) = Integral of (e^0.5x e^x ) from - infinity to 0

e^1.5x dx

= (e^1.5x) / 1.5

= 1/ 1.5

= 2/3

Hope this helps.