Question

Statistics and Probability 10. f(x) = 2w e-2x + 3(1-we-35 with w-U(0,1). Calculate: a. EX) b. VOX ProX > 1

Answer 1

$f(x)=2we^{-2x}+3(1-w)e^{-3x}$

$f_X(x)=\displaystyle\intop^1_0(2we^{-2x}+3(1-w)e^{-3x})dw=e^{-2x}-\frac{3e^{-3x}}{2}$

a.

$E(X)=\displaystyle\intop^{\infin}_{0}x(e^{-2x}-\frac{3e^{-3x}}{2})dx=$

$=(-\frac{(2x+1)e^{-2x}}{4}+\frac{(3x+1)e^{-3x}}{6})|^{\infin}_{0}=\frac{1}{12}$

b.

$V(X)=\displaystyle\intop^{\infin}_{0}x^2(e^{-2x}-\frac{3e^{-3x}}{2})dx-(E(X))^2=$

$=(-\frac{(2x^2+2x+1)e^{-2x}}{4}+\frac{(9x^2+6x+2)e^{-3x}}{18})|^{\infin}_{0}-\frac{1}{144}=\frac{5}{36}-\frac{1}{144}=\frac{19}{144}$

c.

$P(X>1)=\displaystyle\intop^{\infin}_{1}(e^{-2x}-\frac{3e^{-3x}}{2})dx=$

$=(-\frac{e^{-2x}}{2}+\frac{e^{-3x}}{2})|^{\infin}_1=\frac{e^{-2}}{2}-\frac{e^{-3}}{2}=0.0428$