Calculate the variance, 𝑉𝑎𝑟 (𝑋) = 𝜎 2 = ∫ (𝑥 − 𝜇) ∞ −∞ 2 𝑝(𝑥)𝑑𝑥, and standard deviation 𝜎, of a continuous random variable.

A continuous random variable \(X\) has a probability density function defined on the interval \([0, 4]\): \[p(x) = \begin{cases} \frac{3}{32}(4 - x) & \text{if } 0 \le x \le 4 \\ 0 & \text{otherwise} \end{cases}\] a) Calculate the mean \(\mu = E(X)\) of the random variable \(X\). [1 mark] b) Calculate the variance \(\text{Var}(X)\) of the random variable \(X\). [1 mark] c) Hence, calculate the standard deviation \(\sigma = \sqrt{\text{Var}(X)}\) correct to 3 significant figures. [1 mark]

A continuous random variable $X$ has probability density function $p(x) = \begin{cases} \frac{3}{32}(4 - x^2) & \text{if } 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}$ (a) Calculate the mean $\mu$ of $X$. (1 mark) (b) Calculate the variance $\mathrm{Var}(X)$. (1 mark) (c) Hence, calculate the standard deviation $\sigma$ of $X$, giving your answer to 2 decimal places. (1 mark)