Question

If two random variables \[x\] and \[y\] of a bivariate distribution are connected by the relationship \[3x+2y=4\], then correlation coefficient \[{{r}_{xy}}=\]
(1) \[1\]
(2) \[-1\]
(3) \[\dfrac{2}{3}\]
(4) \[-\dfrac{2}{3}\]

Answer 1

Hint: We are given two random variables of a bivariate distribution which are connected by the relationship \[3x+2y=4\]and we are asked to find the correlation coefficient. We will first find the slope of the bivariate distribution using the given relationship. Then, based on the value of the slope, we can determine the correlation coefficient. Hence, we will have the value of the correlation coefficient.

Complete step by step answer:
According to the given question, we are given two random variables of a bivariate distribution which are connected by the relationship \[3x+2y=4\] and using the given information, we have to find the correlation coefficient.
Correlation coefficient depicts the correlation between variables which indicates that as one variable changes in value, the other variable tends to change in a certain specific direction.
For example – height and weight are correlated that is, as the height increases the weight also increases.
We will first find the slope of the given relationship between the bivariate distribution.
The equation that we have is,
\[3x+2y=4\]
Rearranging the above equation in terms of ‘y’, we have the expression as,
\[\begin{align}
  & \Rightarrow 2y=4-3x \\
 & \Rightarrow 2y=-3x+4 \\
 & \Rightarrow y=\dfrac{-3}{2}x+2 \\
\end{align}\]
Comparing the above equation with the standard linear equation, \[y=mx+c\]
We can observe that the slope is negative, that is, the variables are in an inverse relationship.
Therefore, the correlation coefficient is \[-1\].

So, the correct answer is “Option 2”.

Note: We found that the two variables ‘x’ and ‘y’ are inversely correlated. That means if the value of ‘x’ increases then the value of ‘y’ decreases. Also, since the sign of slope is negative we got the direction of correlation between the two variables as inverse \[\left( {{r}_{xy}}=-1 \right)\].