Question

大学数学概率论问题？

大学数学概率论问题求解析

Answer 1

E(X)
=∫(0->无穷) ∫(0->1) 2x^2.e^(-y) dx dy
=(2/3)∫(0->无穷) e^(-y) dy
=-(2/3)[ e^(-y)]|(0->无穷)
=2/3
E(X^2)
=∫(0->无穷) ∫(0->1) 2x^3.e^(-y) dx dy
=(1/2)∫(0->无穷) e^(-y) dy
=-(1/2)[ e^(-y)]|(0->无穷)
=1/2
D(X)=E(X^2) -[E(X)]^2 =1/2 - 4/9 = (9-8)/18=1/18
//
∫(0->无穷) 2xy.e^(-y) dy
=-∫(0->无穷) 2xy de^(-y)
=-2[xye^(-y)]| (0->无穷) +∫(0->无穷) 2x e^(-y) dy
=0-2x[e^(-y)]|(0->无穷)
=2x
E(Y)

=∫(0->1) ∫(0->无穷) 2xy.e^(-y) dy dx
=∫(0->1) 2x dx
=1
//
∫(0->无穷) 2xy^2.e^(-y) dy
= -∫(0->无穷) 2xy^2 de^(-y)
=-2x[y^2.e^(-y)]|(0->无穷) +2∫(0->无穷) 2xye^(-y) dy
=0 +2(2x)
=4x
E(Y^2)

=∫(0->1) ∫(0->无穷) 2xy^2.e^(-y) dy dx
=∫(0->1) 4x dx
=2
D(Y)=E(Y^2) -[E(Y)]^2 =2 - 1 =1

D(X-2Y)=D(X)+4D(Y)=1/18 +4 = 73/8