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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.

Verified Answer :

LammettHash

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

you have

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

which means

\(c=\dfrac3{2^{5/2}}=\dfrac3{4\sqrt2}\)

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