Question

. If \[x,y\] are independent variable , then
A.\[Cov\left( {x,y} \right) = 1\]
B.\[{r_{xy}} = 0\]
C.\[{r_{xy}} = 1\]
D.\[Cov\left( {x,y} \right) = 0\]

Answer 1

Hint: In the given question we have given with Covariance \[Cov\left( {X,Y} \right)\] and Coefficient of correlation \[{r_{XY}}\] , so we have to solve for both the terms . We will use theorems like \[E\left( {XY} \right) = E\left( X \right)E\left( Y \right)\] , where \[E\] is the expected value or mean value and \[X\& Y\] are independent variables. Another theorem will be \[Cov\left( {X,Y} \right) = E\left( {XY} \right) - E\left( X \right)E\left( Y \right)\] , where \[Cov\] is the covariance and \[X\& Y\] are independent variables

Complete step-by-step answer:
Mean - The expected value of \[X\], where \[X\] is a discrete random variable and is a weighted average of all the possible values that \[X\] can take , each value being calculated according to the probability of that event occurring. The expected value of \[X\] is written as \[E\left( X \right)\] .
Covariance is a measure of the association or dependency between two random variables \[X\& Y\] . Covariance can be positive or negative both.
Now , using the above theorems we get ,
\[Cov\left( {X,Y} \right) = E\left( {XY} \right) - E\left( X \right)E\left( Y \right)................\left( i \right)\]
Now putting the value of \[E\left( {XY} \right)\] from first theorem we get in eqn (i) we get ,
\[Cov\left( {X,Y} \right) = E\left( X \right)E\left( Y \right) - E\left( X \right)E\left( Y \right)\]
On solving we get ,
\[Cov\left( {X,Y} \right) = 0............\left( {ii} \right)\] .
Now for Coefficient of correlation \[{r_{XY}}\] , we have a formula
\[{r_{XY}} = \dfrac{{Cov\left( {X,Y} \right)}}{{\sqrt {V\left( X \right)} \sqrt {V\left( Y \right)} }}\], where \[V\left( X \right)\] and \[V\left( Y \right)\] is the variance of \[X\& Y\] respectively .
Now from equation (ii) we have , \[Cov\left( {X,Y} \right) = 0\] , therefore putting this value in above equation we get ,
\[{r_{XY}} = \dfrac{0}{{\sqrt {V\left( X \right)} \sqrt {V\left( Y \right)} }}\]
On solving we get ,
\[{r_{XY}} = 0\]
So, the correct answer is “Option B and D”.

Note: Always check whether the given variables are independent or not. The term covariance is different with variance as the covariance can either be positive or negative but the variance will always be positive . You must remember the results of the theorem only instead of their proofs.