Finding b so that f is probability density function

Question

Find b so that f(x) is probability density function. $f(x)= rect(x-\frac{1}{2})\frac{b}{\sqrt{x}}$

The rules for probability density function are:

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

But I dont know how to handle the rectangle function in these calculations?

Answer 1

1. The presence of the rectangle function means $f(x)$ is only non-zero over a specific interval, and the rectangle function is constant ($1$) over that non-zero interval. Try integrating over that interval rather than over the entire real line.

2. The value of $b$ will come out of the second "rule" you've laid out once you integrate over that finite domain.

Hope this helps!