How do I tell if this function is a probability density function?

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If I have

$$f(x)=\sin(x\pi/10)\qquad\text{for}\;0\leq x\leq10.$$

How do I tell if it is a probability density function? And if it isn't how do I normalize it?

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2 Answers

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You need to check two conditions:

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The integral of a pdf must be equal to one:

$$\int_{-\infty}^{\infty} f(x) \, dx =1$$

In this case, since the function $g(x)=\sin(\pi x/10)$ is defined in $ 0\leq x \leq 10$ and $g(x) \geq 0$ in this interval:

$$\int_{-\infty}^\infty g(x) \, dx = \int_0^{10} \sin(\pi x /10)= \frac{20}{\pi}$$

Then, we scale function $g(x)$ with the inverse of this value and the fdp would be:

$$f(x)=\frac{\pi}{20} \sin \left( \frac{\pi x}{10} \right)$$

Hope this helps!

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