Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]?

Question

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]?

$E[e^{\frac{2X}{3}} - 3] = \int_0^2 \! e^{\frac{2X}{3}} - 3 \, \mathrm{d}x$ $= \frac{3}{2}(e^{\frac{4}{3}} - 5) = -1.8095$

I am integrating over the domain from 0 to 2. But my answer is twice what it should be.

Answer 1

For a continuous random variable $X$ with density $f$ on $[a,b]$, $\mathbb E[g(X)] = \int_a^b g(x) f(x)\; dx$. You forgot to include the density in your integral.