Final answer:
To find the pdf of a continuous random variable, differentiate its cdf. To determine the mean and variance, integrate xf(x) and x²f(x) respectively, then compute variance with Var(X) = E[X²] - (E[X])².
Step-by-step explanation:
The question involves finding the probability density function (pdf) of a continuous random variable X from its cumulative distribution function (cdf), and then determining the mean and variance of X. Given a cdf of F(x), the pdf f(x) is obtained by differentiating F(x) with respect to x.
For the range 1≤x≤2, the pdf will be the derivative of -2x³ + 9x² - 12x - 19, resulting in -6x² + 18x - 12. Outside this range, as described by 'otherwise', the pdf is given as -2x². To find the mean and variance of X, we would need to calculate the expected value E[X] by integrating xf(x), and the expected value of X² by integrating x²f(x), then use these to compute the variance given by Var(X) = E[X²] - (E[X])².
It is important to remember that the pdf must integrate to 1 over the domain of X, and that the mean and variance require integrating over the entire range where f(x) is defined.