Question

Establish for which constants c the following function is a density.

f(x = cx^1/2 on (0,2 and 0 otherwise.
please show the steps taken.

Answer 1

By "density" I assume you mean "probability density function". For this to be the case for
`f(x)`, we require


`\displaystyle\int_(-\infty)^\infty f(x)\,\mathrm dx=1`

Since


`f(x)=\begin{cases}cx^(1/2)&\text{for }0<x<2\\0&\text{otherwise}\end{cases}`

you have


`\displaystyle\int_(-\infty)^\infty f(x)\,\mathrm dx=\int_0^2cx^(1/2)\,\mathrm dx=\frac{2c}3x^(3/2)\bigg|_(x=0)^(x=2)=\frac{2^(5/2)c}3=1`

which means


`c=\frac3{2^(5/2)}=\frac3{4\sqrt2}`