Let [math]X[/math] be a random variable with range [math][-1,1][/math] and density function [math]f_X(x) = ax^2 + bx + c[/math] if [math]|x| \lt 1[/math] and 0 otherwise.

• Show that [math]2a/3 + 2c = 1[/math] (see Exercise).
• Show that [math]2b/3 = \mu(X)[/math].
• Show that [math]2a/5 + 2c/3 = \sigma^2(X)[/math].
• Find [math]a[/math], [math]b[/math], and [math]c[/math] if [math]\mu(X) = 0[/math], [math]\sigma^2(X) = 1/15[/math], and sketch the graph of [math]f_X[/math].
• Find [math]a[/math], [math]b[/math], and [math]c[/math] if [math]\mu(X) = 0[/math], [math]\sigma^2(X) = 1/2[/math], and sketch the graph of [math]f_X[/math].