Question

The density function of x is given by f(x) = a * b * x²⁰ if 0 ≤ x ≤ 1, otherwise. If the expectation of x is E(x) = 0.75, find the values of a and b.

Answer 1

To determine the values of constants 'a' and 'b' in the given PDF, one must solve two equations: one for the normalization of the PDF (integral equals 1) and the other for the expected value (integral of the PDF times x equals 0.75).

Step-by-step explanation:

The question asks to find the values of constants a and b in a probability density function (PDF) given that the expectation, or mean, of x is 0.75. The density function is f(x) = a * b * x²⁰ for the interval [0,1]. To find a and b, we need to satisfy two conditions. First, the integral of the PDF over its range should equal 1, for it to be a valid PDF. Second, the expected value E(x) must be equal to 0.75.

To normalize the PDF, we set the integral from 0 to 1 of f(x) equal to 1:

∫01 a * b * x²⁰ dx = 1.

Solving this equation will give us the product a * b. Next, to find the expectation, we calculate:

E(x) = ∫01 x * a * b * x²⁰ dx.

This should equal 0.75. By solving these two equations, we can find the appropriate values for a and b that satisfy the given conditions.