Questions: The proportion of people who respond to a certain mail-order solicitation is a random variable X having the following density function. f(x)=2(x+1)/3, 0<x<1 0, elsewhere Find σg(X)² for the function g(X)=2X²+3. σg(X)²= (Round to three decimal places as needed.)

Solution Steps

To find the variance of \( g(X) \), we need to follow these steps:

1. Compute the expected value \( E[g(X)] \).
2. Compute the expected value \( E[g(X)^2] \).
3. Use the formula for variance: \( \sigma_{g(X)}^2 = E[g(X)^2] - (E[g(X)])^2 \).

Given the density function: \[ f(x) = \begin{cases} \frac{2(x+1)}{3}, & 0 < x < 1 \\ 0, & \text{elsewhere} \end{cases} \] and the function \( g(X) = 2X^2 + 3 \).

The expected value \( E[g(X)] \) is calculated as: \[ E[g(X)] = \int_{0}^{1} g(x) f(x) \, dx = \int_{0}^{1} (2x^2 + 3) \left( \frac{2(x+1)}{3} \right) \, dx \] Evaluating this integral, we get: \[ E[g(X)] = \frac{34}{9} \]

First, compute \( g(X)^2 \): \[ g(X)^2 = (2X^2 + 3)^2 = 4X^4 + 12X^2 + 9 \] Then, compute the expected value \( E[g(X)^2] \): \[ E[g(X)^2] = \int_{0}^{1} g(x)^2 f(x) \, dx = \int_{0}^{1} (4x^4 + 12x^2 + 9) \left( \frac{2(x+1)}{3} \right) \, dx \] Evaluating this integral, we get: \[ E[g(X)^2] = \frac{659}{45} \]

Using the formula for variance: \[ \sigma_{g(X)}^2 = E[g(X)^2] - (E[g(X)])^2 \] Substitute the values: \[ \sigma_{g(X)}^2 = \frac{659}{45} - \left( \frac{34}{9} \right)^2 \] Simplify the expression: \[ \sigma_{g(X)}^2 = \frac{151}{405} \] Convert to decimal and round to three decimal places: \[ \sigma_{g(X)}^2 \approx 0.373 \]

Final Answer