Answer in Statistics and Probability for aditi #249600
Let f(x1,x2,x3)=exp[−(x1 + x2 + x3)], 0 < x1 < ∞, 0 < x2 < ∞, 0 < x3 < ∞, = zero elsewhere, be the joint pdf of X1, X2, X3. (a) Compute P (X1 < 2X2) and P (X1 = 2X2). (b) Determine the mgf of X1+2X2+X3. Are X1, X2, and X3 random variables independent? Give reasoning.
a)
“P(X_1<2X_2)=int^{infin}_0int^{infin}_{0}int^{2x_2}_0e^{-(x_1+x_2+x_3)}dx_1dx_2dx_3=”
“=-int^{infin}_0int^{infin}_{0}(e^{-(3x_2+x_3)}-e^{-(x_2+x_3)})dx_2dx_3=”
“=-int^{infin}_{0}(-e^{-(3x_2+x_3)}/3+e^{-(x_2+x_3)})|^{infin}_0dx_3=”
“=-int^{infin}_{0}(e^{-x_3}/3-e^{-x_3})dx_3=1-1/3=2/3”
“P(X_1=2X_2)=int^{infin}_0int^{infin}_{0}int^{2x_2}_{2x_2}e^{-(x_1+x_2+x_3)}dx_1dx_2dx_3=0”
mgf of X1+2X2+X3:
“M_{X_1+2X_2+X_3}(t)=int^{infin}_0int^{infin}_0int^{infin}_0e^{(t-1)(x_1+2x_2+x_3)}dx_1dx_2dx_3=infin”
X1, X2,X3 are independent if the joint p.d.f. is the product of the individual p.d.f.’s
We have:
“f_{X_1}=f_{X_2}=f_{X_3}=-e^{-(x_1+x_2+x_3)}”
“f_{X_1}cdot f_{X_2}cdot f_{X_3}=-e^{-3(x_1+x_2+x_3)}neq f(X_1, X_2,X_3)”
So, the variables are dependent.
