Let [math]X[/math] be a random variable with range [math][-1,1][/math] and density function [math]f_X(x) = ax + b[/math] if [math]|x| \lt 1[/math].

• Show that if [math]\int_{-1}^{+1} f_X(x)\, dx = 1[/math], then [math]b = 1/2[/math].
• Show that if [math]f_X(x) \geq 0[/math], then [math]-1/2 \leq a \leq 1/2[/math].
• Show that [math]\mu = (2/3)a[/math], and hence that [math]-1/3 \leq \mu \leq 1/3[/math].
• Show that [math]\sigma^2(X) = (2/3)b - (4/9)a^2 = 1/3 - (4/9)a^2[/math].