Question

The probability density function of a random variable X is $P_x\left(x\right)=e^{-x}\text{for}x\ge 0\text{and}0\text{otherwise.}$ The expected value of the function $g_x\left(x\right)=e^{3x/4}\text{is}$ .

• A. 4

Answer 1

The correct option is A 4

We know that if f(x) is the p.d.f. for Random variable x then $E\left(x\right)={\int }_{-\infty }^{\infty }xf\left(x\right)dx$

Now if g(x) is any other function of 'x' then

$E\left(g\right(x\left)\right)={\int }_{-\infty }^{\infty }g\left(x\right).f\left(x\right)dx$

So $E\left(e^{3x/4}\right)={\int }_{-\infty }^{\infty }{e}^{3x/4}.P_x\left(x\right)dx$

$={\int }_0^{\infty }{e}^{3x/4}.e^{-x}dx={\int }_0^{\infty }{e}^{-x/4}dx=4$