4. Incorrect. The answer is true, not false.

The probability function of X is P(x) = 1/6, x = 1,2,3,4,5,6. We know that E[H(X)] = ∑ H(x)p(x) where H(X) is a function of X. In this particular case, H(X) = X². First of all, we have to calculate E[X²] in order to calculate E[2X² - 5].

E[X²] = 1²*(1/6) + 2²*(1/6) + 3²*(1/6) + 4²*(1/6) + 5²*(1/6) + 6²*(1/6) = 91/6

Then, by applying some theorems on expectations, we have that:

E[2X² - 5] = 2E(X²) - 5 = 2*(91/6) - 5 = 76/3 ≈ 25 of US$ 2,500.