If the probability density function of a continuous random variable X is `f(x)={{:((3+2x)/(18),2lexle4),(0,xlt2" or"xgt4):}` Then the mathematical expectation of X is

Options

A. `(83)/(27)`
B. `(27)/(83)`
C. `(87)/(23)`
D. `(38)/(72)`

Answer

A

Explanation

`E(X)=oversetoounderset(-oo)intxf(x)dx=0+overset4underset2intx((3+2x)/(18))dx+0`
`=1/(18)overset4underset2int(3x+2x^2)dx=1/(18)[(3x^2)/2+(2x^3)/3]_2^4`
`=1/(18)[(24+(128)/3)-(6+(16)/3)]=1/(18)xx(166)/3=(83)/(27)`