Final answer:
To determine the pdf, take the derivative of the cdf. The mean of X is -7/9, and the variance of X is 56/81.
Step-by-step explanation:
To determine the probability density function (pdf), f(x), we need to calculate the derivative of the cumulative distribution function (cdf), F(x). In this case, the cdf is given by -2x + 9x³ - 12x - 19 for 1≤x≤2 and 0 otherwise.
So, by taking the derivative, we get f(x) = dF(x)/dx = -2 + 27x² - 12.
To find the mean, we need to calculate the expected value of X, which is given by E(X) = ∫[1,2] x*f(x) dx. By plugging in the values, we can find that E(X) = -7/9. To find the variance, we need to calculate E(X²) - (E(X))². By evaluating the integral E(X²) = ∫[1,2] x²*f(x) dx, we find that E(X²) = -1/9. Therefore, the variance is Var(X) = -1/9 - (-7/9)² = 56/81.