Consider two exponentially distributed random variables X and Y, both having a mean of 0.50. Let Z = X + Y and r be the correlation coefficient between X and Y. If the variance of Z equals 0, then the value of r is _______ (round off to 2 decimal places).
Let λ be the parameter of exponential distribution
So, for exponentially distributed random variables X & Y
Expected Value or Mean \(E\left( X \right) = \frac{1}{\lambda }\)
Variance: \(Var\;\left( X \right) = \frac{1}{{{\lambda ^2}}}\)
Var (X + Y) = Var X + Var Y + 2 Cov (X, Y)
Where Cov (X, Y) = Covariance of X and Y
Correction coefficient is given by:
\(r = \frac{{Cov\;\left( {X,\;Y} \right)}}{{\sqrt {Var\left( X \right)Var\left( Y \right)} }}\)
Calculation:
Given: E(X) = E(Y) = 0.5
\(\frac{1}{{{\lambda _1}}} = \frac{1}{{{\lambda _2}}} = 0.5\)
λ1 = λ2 = 2
\(Var\;\left( X \right) = \frac{1}{{\lambda _1^2}} = \frac{1}{{{2^2}}} = 0.25\)
\(Var\;\left( Y \right) = \frac{1}{{\lambda _2^2}} = \frac{1}{{{2^2}}} = 0.25\)
Also it is given, Z = X + Y
And Var (Z) = 0
∴ Var (X + Y) = 0
Var X + Var Y + 2 Cov (X, Y) = 0
On putting values:
0.25 + 0.25 + 2 Cov (X, Y) = 0
\(Cov\;\left( {X,\;Y} \right) = - \frac{{0.5}}{2} = - 0.25\)
Correction coefficient is given by:
\(r = \frac{{Cov\;\left( {X,\;Y} \right)}}{{\sqrt {Var\left( X \right)Var\left( Y \right)} }} = \frac{{ - 0.25}}{{\sqrt {0.25 \times 0.25} }} = \; - 1\)Important Questions from Correlation Analysis
If X ∼ N (0, 1) and Y = X2 then the correlation coefficient r (X, Y) is
Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): If the securities with less than perfect negative correlation between their price movements are combined, portfolio risk can be reduced significantly.
Reason (R): The term with negative correlation has the effect of reducing the computed value of total portfolio risk, given other terms that are positive.
In the light of the above statements, choose the most appropriate answer from the options given below:
The two-regression equation of variable \(\rm{x}\) and \(\rm{y}\) are
\(\rm{y = 0.8x + 9.8}\) and \(\rm{x = 10.2 + 0.6y}\)
The coefficient of correlation between \(\rm{x}\) and \(\rm{y}\) is
- In which conditions, Karl Pearson's correlation coefficient can be calculated ?
A. If means of both the variables are equal
B. If one variable is measured in interval scale and another is measured in ordinal scale
C. If there is linear relationship between two variables
D. If data are obtained in interval or ratio scale for both the variables
E. If direction of relationship between two variables is known
Choose the most appropriate answer from the options given below : The coefficients of the regression β X|y and β Y|x , are known, The coefficient of correlation equals:
