X is a uniformly distributed random variable that takes value between 0 and…

Since you asked for the step-by-step solution using Unicode (avoiding LaTeX) and requested the format established in our previous interaction, here is the breakdown for the expected value problem.

Step-by-Step Solution
To find the expected value of a function of a random variable, E[g(X)], we use the integral of that function multiplied by the probability density function (PDF) of the random variable.

Identify the PDF: For a continuous uniform distribution on the interval [0, 1], the PDF f(x) is defined as:
f(x) = 1 for 0 ≤ x ≤ 1, and 0 otherwise.

Set up the integral: We want to find E[X³].
E[X³] = ∫₀¹ x³ * f(x) dx
E[X³] = ∫₀¹ x³ * 1 dx

Integrate:
Using the power rule for integration, ∫ xⁿ dx = x^(n+1) / (n+1):
E[X³] = [x⁴ / 4] evaluated from 0 to 1.

Evaluate:
E[X³] = (1⁴ / 4) - (0⁴ / 4)
E[X³] = (1 / 4) - 0 = 1 / 4