Calculating the pdf from a given cdf of a CRV
Calculating the pdf from a given cdf of a CRV
IB Year 5 | Grade 11
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 \(y = \mathrm{f}(x)\).
Solution:
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Answer:(a) \(k=\frac{1}{27}\). (b) \(\mathrm{f}(x)=\frac{x^2}{9}\) for \(0\leq x\leq3\), and \(0\) otherwise; sketch shown.
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