设随机变量X的概率密度为
f(x)=
{acosx,∣x∣≤π/2,
0,其他
求：(1)常数a；(2){0﹤X﹤π/4}；(3)X的分布函数F(x).
【正确答案】：(1)∫+∞-∞f(x)dx=∫π/2-(π/2)acosxdx=2∫π/20acosxdx=2a=1 得a=1/2 (2)P{0﹤X﹤π/4}=∫π/10(1/2)cosxdx=√2/4 (3)当x﹤-(π/2)时F(x)=∫x-∞f(t)dt=0 当-(π/2)≤x﹤π/2时F(x)=∫x-(π/2)(1/2)costdt=1/2(1+sinx) 当x≥π/2时F(x)=∫π/2-(π/2)(1/2)costdt=1 ∴X的分布函数为F(x)= {0 x﹤-(π/2) 1/2(1+sinx) -(π/2)≤x﹤π/2 1 x≥π/2
