Probability & Statistics Ch.2 Section 4 Question 4

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Establish for which constants c the following functions are densities:

a) $f(x) = cx$ on $(0, 1)$ and 0 otherwise.
b) $f(x) = cx^n$ on $(0, 1)$ and 0 otherwise, for n a nonnegative integer.
c) $f(x) = cx^{1/2}$ on $(0, 2)$ and 0 otherwise.
d) $f(x) = c \sin(x)$ on $(0, \frac{\pi}{2})$ and 0 otherwise.

Solution

For a function to be a density, its integral must equal 1.

(a) $f(x) = cx$ on $(0, 1)$

This forms a right triangle with base 1 and height c:

\int_0^1 cx \, dx = \frac{cx^2}{2}\Big|_0^1 = \frac{c}{2} = 1

c = 2

(b) $f(x) = cx^n$ on $(0, 1)$

\int_0^1 cx^n \, dx = \frac{cx^{n+1}}{n+1}\Big|_0^1 = \frac{c}{n+1} = 1

c = n + 1

(c) $f(x) = cx^{1/2}$ on $(0, 2)$

\int_0^2 c\sqrt{x} \, dx = \frac{2cx^{3/2}}{3}\Big|_0^2 = \frac{2c(2)^{3/2}}{3} = \frac{2c \cdot 2\sqrt{2}}{3} = \frac{4c\sqrt{2}}{3} = 1

c = \frac{3}{4\sqrt{2}} = \frac{3\sqrt{2}}{8}

(d) $f(x) = c\sin(x)$ on $(0, \frac{\pi}{2})$

\int_0^{\pi/2} c\sin(x) \, dx = -c\cos(x)\Big|_0^{\pi/2} = -c\cos(\frac{\pi}{2}) + c\cos(0) = -c(0) + c(1) = c = 1

c = 1