Chapter 2 Section 4 Question 4
Question
Establish for which constants c the following functions are densities.
  1. $f(x) = cx$ on $(0, 1)$ and 0 otherwise.
  2. $f(x) = cx^n$ on $(0, 1)$ and 0 otherwise, for n a nonnegative integer.
  3. $f(x) = cx^{\frac{1}{2}}$ on $(0, 2)$ and 0 otherwise.
  4. $f(x) =c*sin(x)$ on $(0, \frac{\pi}{2})$ and 0 otherwise.
Answer
Author: Mohammad-Ali Bandzar| Date:Oct 21 2020
(a) $f(x) = cx$ on $(0, 1)$ and 0 otherwise.
This will be a right angle triangle, the area is $$A=\frac{1}{2}*base*height$$ $$1=\frac{1}{2}*1*height$$ $$\frac{1}{\frac{1}{2}}=height$$ $$height=2$$ Therefore c=2
(b) $f(x) = cx^n$ on $(0, 1)$ and 0 otherwise, for n a nonnegative integer.
$$\int cx^n dx=\frac{cx^{n+1}}{n+1} +c$$ $$\int_{0}^{1} cx^n dx=\frac{c*1^{n+1}}{n+1}-\frac{c*0^{n+1}}{n+1}$$ $$\int_{0}^{1} cx^n dx=\frac{c}{n+1}$$ $$1=\frac{c}{n+1}$$ $$c=n+1$$
(c) $f(x) = cx^{\frac{1}{2}}$ on $(0, 2)$ and 0 otherwise.
$$\int c\sqrt{x} dx=\frac{c*2x^\frac{3}{2}}{3}$$ $$\int_{0}^{2} c\sqrt{x} dx=\frac{c*2(2)^\frac{3}{2}}{3}-\frac{c*2(0)^\frac{3}{2}}{3}$$ $$\int_{0}^{2} c\sqrt{x} dx=\frac{c*2(2)^\frac{3}{2}}{3}$$ $$1=\frac{c*2(2)^\frac{3}{2}}{3}$$ $$3=c*2(2)^\frac{3}{2}$$ $$\frac{3}{2}=c(2)^\frac{3}{2}$$ $$c=\frac{3}{4\sqrt{2}}$$
(d) $f(x) =c*sin(x)$ on $(0, \frac{\pi}{2})$ and 0 otherwise.
$$f'(x)=-c*cos(x)$$ $$\int_{0}^{\frac{\pi}{2}} c*sin(x) dx = F(\frac{\pi}{2})-F(0)$$ $$F(\frac{\pi}{2})-F(0)=-c*cos(\frac{\pi}{2})-(-c*cos(0))$$ $$1=-c*cos(\frac{\pi}{2})+c*cos(0)$$ since $cos(0)=1$ and $cos(\frac{\pi}{2})=0$ $$1=c$$ $$c=1$$