Question

设随机变量X的分布函数为F(X)，概率密度f(x)=af1(x)+bf2(x)

其中f1(x)是正态分布N(0,1)的概率密度，f2(x)是在[0,2]上服从均匀分布的随机变量的概率密度，且F（0）=1/4，求a，b

Answer 1

F(0) = (1/2)a = 1/4
=> a= 1/2
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F(∞) = a+ b = 1
1/2 +b =1
b=1/2

Answer 2

F（0）= ∫af1(x)dx│(x=-∞ to 0)=a/2=1/4
a=1/2

∫f(x)dx│(x=-∞ to +∞)=∫af1(x)dx│(x=-∞ to +∞)+∫bf2(x)dx│(x=0 to 2)=1/2+b=1
b=1/2

Answer 3

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