Please solve this

Question

Find E[X] when the density function of X is f (x) =2x, 0≤x ≤1 0, otherwise

Answer 1

I'm assuming that the random variable X in the problem is continuous. Recall that expected value of

a continuous random variable is given by:

E[X] = ∫_–∞^∞ xf(x)dx

where f(x) is the density of the cumulative distribution function of X.

In particular, if the density function is defined to be f(x) = 2x if 0≤x≤, and 0 otherwise, then:

E[X] = ∫_–∞^∞ xf(x)dx = ∫₀¹ x(2x)dx = ∫₀¹ 2x²dx = 2(1)³ / 3 - 2(0)³ / 3 = 2/3.