The continuous random variable $X$ has probability density function \[\mathrm{f}(x) = \begin{cases

Question

The continuous random variable $X$ has probability density function $$\mathrm{f}(x) = \begin{cases} 4k & , \quad \frac{2}{3} \leq x < 2 \ kx(4-x) & , \quad 2 \leq x \leq 4 \ 0 & , \quad \text{otherwise} \end{cases}$$ where $k$ is a constant. Find the value of the constant $k$. Find $\mathrm{E}(X)$. Find $\mathrm{E}(X^2)$. Find $\mathrm{Var}(X)$ and the standard deviation of $X$.

Accepted Answer

(a) $\frac{3}{32}$; (b) $2.04$; (c) $4.86$; (d) $0.695$, $0.833$.

Calculation of expectation and variance of a CRV

Calculation of expectation and variance of a CRV