Chapter 2 Section 4 Question 5

Question

Is the function defined by $f (x) =\frac{x}{3}$ for $-1 < x < 2$ and 0 otherwise, a density? Why or why not?

Answer

Author: Mohammad-Ali Bandzar| Date:Oct 21 2020

This forms a right angle triangle, the area(integral) of which will be: $$Area=base*height$$ $$Area=(2--1)*f(2)$$ $$Area=(3)*\frac{2}{3}$$ $$Area=2$$ No this is not a density because the integral of the function is greater than one. This violates textbook definition 2.4.2 which states that for any density function $\int_{-\infty}^{\infty} f(x)dx=1$

This forms a right angle triangle, the area(integral) of which will be: $$Area=base*height$$ $$Area=(2--1)*f(2)$$ $$Area=(3)*\frac{2}{3}$$ $$Area=2$$ No this is not a density because the integral of the function is greater than one. This violates textbook definition 2.4.2 which states that for any density function $\int_{-\infty}^{\infty} f(x)dx=1$