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The given probability density function (PDF) for the Beta-distributed random variable \( X \) with parameters \( a = 3 \) and \( b = 2 \) is: \[ f(x) = \begin{cases} 12x(1 - x), & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} \] We will now compute the following: 1. Cumulative Distribution Function (CDF) 2. \( P(0 < X < 1/2) \) 3. Mean and Variance using integration --- ### (1) CDF of \( X \) The cumulative distribution function (CDF), \( F(x) \), is obtained by integrating the PDF from 0 to \( x \): \[ F(x) = \int_0^x 12t(1 - t) dt \] Expanding the expression inside the integral: \[ F(x) = \int_0^x 12t - 12t^2 dt \] Now, integrating each term separately: \[ F(x) = 12 \int_0^x t dt - 12 \int_0^x t^2 dt \] Using the standard integration formulas: \[ \int t dt _____ ____ _____ _________ _______ _________ ____ ____ _________ _____.
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