The continuous random variable $X$ has cumulative distribution function $F(x)$ where \[F(x) = \b

Question

The continuous random variable $X$ has cumulative distribution function $F(x)$ where $$F(x) = \begin{cases} 0 & , \quad x < 0 \ kx^3 & , \quad 0 \leq x \leq 3 \ 1 & , \quad x \geq 3 \end{cases}$$ Find the value of the constant $k$. Find the probability density function $\mathrm{f}(x)$ and sketch the graph of $y = \mathrm{f}(x)$.

Accepted Answer

(a) $\frac{1}{27}$; (b) $\mathrm{f}(x) = \frac{x^2}{9}, 0 \leq x \leq 3$

Calculating the pdf from a given cdf of a CRV

Calculating the pdf from a given cdf of a CRV