True or False. 5. If f_{X,Y} (x, y)>0 for all x and y, then f_{X,Y} (x, y) = f_{X|Y} (x, y) f_Y (y). 6. For any joint ra

[answerhappygod]

True or False.
5. If f_{X,Y} (x, y)>0 for all x and y,
then f_{X,Y} (x, y) = f_{X|Y} (x, y) f_Y (y).
6. For any joint random discrete variables X, Y, and Z, we
have p_{X,Y,Z}(x, y, z) = p_X(x)p_{Y |X}(y, x) p_{Z|X,Y} (z, x, y)
.
7. When X and Y are jointly continuous random variables,
f_{X|Y} (x, y) = f_X(x)f_{Y|X}(y, x)/f_Y(y) for
f_Y(y)>0.
8. X = Y a.s. iff E[(X − Y )^2 ] = 0
9. Random variables X_1, X_2, ..., X_n are pairwise
independent if f_{X1,X2,...,Xn} (x_1, x_2, ..., x_n) =
\Prod_{i=1}^n f_{X_i} (x_i), ∀xi .
10. If F_{X,Y,Z}(x, y, z) = F_X(x)F_Y (y)F_Z(z), for all x,
y, z, then the random variables X, Y, and Z are independent.
11. If F_{X,Y,Z}(x, y, z) = F_X(x)F_Y (y)F_Z(z) for all x,
y, and z, then the random variables X, Y, and Z are
independent.
12. Jointly distributed random variables X and Y are fully
characterized by G(x_i , x_f ; y_i , y_f ) \triangleq P({s : x_i
< X(s) ≤ x_f } ∩ {s : y_i < Y (s) ≤ y_f }).
13. Given two jointly distributed random variables X and Y
, if F_X(x) and F_Y (y) are known, then F_{X,Y} (x, y) can be
uniquely determined.
14. F_X(x) = \int_{-\infty}^\infty f_{X,Y} (x, y)dx.
15. f_X(x) = f_{X,Y} (x, ∞).
16. Given a probability space (S, A, P), an n-dimensional
random vector X is a mapping from a sample space to a subset of R^n
.
17. If Z = X + Y , then f_Z(z) = f_X(z) ∗ f_Y (z).
18. If X and Y are independent random variables, then the
pdf of Z = X + Y is f_Z(z) = f_X(z) ∗ f_Y (z).