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

Question

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?

Answer 1

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!

Answer 2

You need to check two conditions,

1)- $ f(x) $ has to be non-negative,

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