Question

(1)设ξ是离散型随机变量，η=3ξ+2,求证：Eη=3Eξ+2,Dη=9Dξ;

(2)对于上述问题能否推广到一般的离散型随机变量间线性关系的数学期望及方差的关系式?并证明你的结论.

Answer 1

（1）证明:令P(ξ=i)=p_i.

Eη=（3×i₁+2)+(3×i₂+2)+…+(3×i_n+2)??

=3×(i₁+i₂+…+i_n)+2(++…+)

=3Eξ+2.?

Dη=［3×i_i+2-(3Eξ+2)］²×+［3×i₂+2-(3Eξ+2)］²×+…+［3×i_n+2-(3Eξ+2)］²×??

=9［(i₁-Eξ)²+(i₂-Eξ)²+…+(i_n-Eξ)²］=9Dξ.?

(2)解析:易猜得η=aξ+b时，Eη=aξ+b，Dη=a²Dξ.?

证明过程同（1）.