91Ó°ÊÓ

The Cumulative Distribution Function (CDF) provides the probability that a random variable takes on a value less than or equal to a given number. For the given newspaper advertising problem, the beta probability density function is: \[f(x) = 30 x^2 (1-x)^2, \text{ for } 0 \leq x \leq 1. \]To find the CDF, we need to integrate this density function from 0 to a variable \(x\): \[F(x) = \int_0^x 30 t^2 (1-t)^2 dt. \]This integral gives us the cumulative probability up to any value \(x\). First, we need to expand the integrand and then integrate each term: \[F(x) = \int_0^x \(30(t^2 - 2t^3 + t^4)\) dt. \] After integrating, we get: \[F(x) = 10x^3 - 15x^4 + 6x^5. \]So the CDF helps us understand the accumulation of probabilities across the values of \(x\).

The Expected Value (E(X)) represents the average or mean value of a random variable if the experiment is repeated many times. For a Beta distribution \(\text{Beta}(\alpha,\beta)\), the formula to calculate the expected value is: \[E(X) = \frac{\alpha}{\alpha + \beta}. \]In our problem, both \( \alpha \) and \( \beta \) are 3, so the expected value is: \[E(X) = \frac{3}{3+3} = \frac{1}{2}. \]This means that, on average, 50% of the newspaper’s space is expected to be devoted to advertising. This average or mean extends to the concept that if we were to repeatedly analyze the newspaper over numerous days, we would find that about half of each issue is dedicated to ads.

Variance summarizes how much the values of the random variable deviate from the expected value. For a Beta distribution, the variance is computed using the formula: \[ \operatorname{Var}(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}. \]With \( \alpha = 3 \) and \( \beta = 3 \): \[ \operatorname{Var}(X) = \frac{3 \cdot 3}{(3 + 3)^2 (3 + 3 + 1)} = \frac{9}{(6)^2 \cdot 7} = \frac{1}{28}. \]This tells us how spread out the advertising proportions are around the mean (0.5). A lower variance indicates that the proportions tend to be closer to the mean, indicating less variability in the share of advertising in the newspaper.

Integrating polynomials is a critical step in finding the CDF of a Beta distribution. In our case, we start with the polynomial \( 30 t^2 (1 - t)^2 \). First, expand it: \[ 30 t^2 (1 - t)^2 = 30 (t^2 - 2t^3 + t^4). \]Next, integrate each term separately: \[30 \int_0^x (t^2 - 2t^3 + t^4) dt = 30 \left[ \frac{t^3}{3} - \frac{2t^4}{4} + \frac{t^5}{5} \right]_0^x = 30 \left( \frac{x^3}{3} - \frac{x^4}{2} + \frac{x^5}{5} \right). \]Simplified, we have: \[F(x) = 10x^3 - 15x^4 + 6x^5. \]Understanding polynomial integration is crucial as it allows us to find the exact expressions needed for the CDF in probability problems.

The Beta Distribution is very flexible and useful for modeling random variables limited to an interval \([0, 1]\). It is defined with two parameters, \(\alpha\) and \(\beta\), which shape the distribution. The probability density function (PDF) for a Beta distribution is: \[f(x | \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \]where \( B(\alpha, \beta) \) is the Beta function. In our example: \( \alpha = 3 \) and \( \beta = 3 \) transform the beta function to: \[ f(x) = 30 x^2 (1-x)^2. \]This distribution is symmetrical because \( \alpha = \beta \), modeling a situation where the proportion of advertising can range anywhere within 0 and 1, but mostly clusters around the mean (0.5). The Beta distribution’s flexibility and bounded nature make it ideal for proportions and probabilities.