2023-11-15
Let’s say we have a pdf, \(f_X(x) = \dfrac{1}{9}x^2\) for \(0 \leq x \leq 3\)
If we only got information that \(f_X(x)\) is defined for \(0 \leq x \leq b\). What is \(b\) to make the pdf \(f_X(x) = \dfrac{1}{9}x^2\) valid?
\[\int_0^bf_X(x) =1\]
\[\int_0^b\dfrac{1}{9}x^2 = 1\] \[\dfrac{1}{27} x^3 \Big|_{x=0}^{x=b}= 1\] \[\dfrac{1}{27}b^3 - \dfrac{1}{27}0^3 = 1\]
Then \(b\) must be…
\[\dfrac{1}{27}b^3= 1\] \[b^3 = 27\] \[b = 3\]
