Standard +0.3 This is a standard continuous probability distribution question requiring routine integration to find k, then E(X) and Var(X). The symmetry of the quadratic function allows E(X) to be stated immediately (midpoint = 3), and the variance calculation follows standard formulas with straightforward polynomial integration. Slightly easier than average due to the symmetric property and mechanical nature of the calculations.
The probability density function, f, of a random variable \(X\) is given by
$$\text{f}(x) = \begin{cases} k(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise,} \end{cases}$$
where \(k\) is a constant.
State the value of \(\text{E}(X)\) and show that \(\text{Var}(X) = \frac{9}{5}\). [6]
The probability density function, f, of a random variable $X$ is given by
$$\text{f}(x) = \begin{cases} k(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise,} \end{cases}$$
where $k$ is a constant.
State the value of $\text{E}(X)$ and show that $\text{Var}(X) = \frac{9}{5}$. [6]
\hfill \mbox{\textit{CAIE S2 2021 Q6 [6]}}