Question

该题是出自《概率论与数理统计》，请问有谁知道24题的详细解法！！

Answer 1

f(x,y)=(1/2)(x+y)e^(-(x+y)

(1)
f(x)=∫[0到∞](1/2)(x+y)e^(-(x+y))dy
= (1/2)∫[0到∞]xe^(-(x+y))dy + (1/2)∫[0到∞]e^(-(x+y))dy
= (1/2)xe^(-x)∫[0到∞]e^(-y)dy + (1/2)e^(-x)∫[0到∞]ye^(-y)dy
= (1/2)xe^(-x)+ (1/2)e^(-x)
= (1/2)(1+x)e^(-x), x>0; =0 其余.
同理,
f(y) = (1/2)(1+y)e^(-y), y>0; =0 其余.
f(x,y) ≠ f(x)f(y), 所以 X 和 Y 不独立.

(2)
f(z) = ∫f(z-y,y)dy
= ∫[0,∞](1/2)(x+y)e^(-(x+y))dy
= ∫[0,∞](1/2)(z-y+y)e^(-(z-y+y))dy
= ∫[0,z](1/2)ze^(-z)dy
= (1/2)z²e^(-z), z>0; =0, 其余.