Question

Given is a random variable x with probability density function f given by f(x) = 0 for x < 0, and for x > 1, and f(x)=4x − 4x3 for 0 ≤ x ≤ 1. determine the expectation and variance of the random variable 2x 3.

Answer 1

To determine the expectation and variance of the random variable 2x^3, first find its probability density function (PDF) by substituting y = 2x^3 into the given PDF for x. Then, use the formulas for expectation and variance to calculate the desired values.

Step-by-step explanation:

To find the expectation and variance of the random variable 2x^3, we first need to find its probability density function (PDF) and then use the formulas for expectation and variance. The given PDF for x is f(x)= 4x - 4x^3 for 0 ≤ x ≤ 1, and 0 otherwise. To find the PDF of 2x^3, we substitute y = 2x^3 into f(x) and solve for f(y). Then, we calculate the expectation and variance using the new PDF.

First, let's find the PDF of y. Substituting y = 2x^3 into f(x), we get f(y) = 4(2x^3) - 4(2x^3)^3 = 8x^3 - 128x^9 for 0 ≤ x ≤ 1, and 0 otherwise.

Now, we can calculate the expectation and variance. The expectation of a random variable is given by E(y) = ∫(y * f(y)) dy, and the variance is given by Var(y) = ∫((y - E(y))^2 * f(y)) dy. Plugging in the PDF f(y) = 8x^3 - 128x^9 and evaluating the integrals within the limits 0 to 1, we can find the expectation and variance of the random variable 2x^3.