Calculate the values of the constants a and b for the probability density function f(x) = ax(b−x2) for 0 x 1.

a) Calculate the values of the constants $a$ and $b$

Step 1: For a probability density function (PDF), the total probability over its domain must be 1.

$$\int_{-\infty}^{\infty} f(x)\,dx = 1$$

Given $f(x) = ax(b-x^2) = abx - ax^3$ for $0 \le x \le 1$.

$$\int_{0}^{1} (abx - ax^3)\,dx = 1$$ $$\left[ \frac{abx^2}{2} - \frac{ax^4}{4} \right]_{0}^{1} = 1$$ $$$$