Two random variable X and Y are totally uncorrelated if ρ XY :
| Two random variable X and Y are totally uncorrelated if ρXY:
A. 1.2
B. <span style="font-family:times new roman,times,serif;">∞</span>
C. Negative
D. 0
Please scroll down to see the correct answer and solution guide.
Right Answer is: D
SOLUTION
Concept:
Correlation Coefficient (ρ) of two Random Variables x and y is given by:
\({\rho _{xy}} = \frac{{{C_{ov}}\left[ {x,\;y} \right]}}{{\sqrt {{V_{ar}}\left[ x \right]{v_{ar}}\left[ y \right]} }}\) ---(1)
Where,
Cov [x, y] = E [X, Y] – E [X] E[Y] ---(2)
Analysis:
Given X & Y are totally uncorrelated, i.e.
E[X Y] = E[X] E[Y]
Substituting this is Equation (2), we get:
Cov [X, Y] = E[X Y] – E [X] E [Y] = 0
∴ Cov [X, Y] = 0
Hence from eqn (1), ρxy = 0