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
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:
X, Y and Z are three uncorrelated variables having variances \(\sigma_x^2, \sigma_y^2 \:and\:\sigma_z^2\) respectively, then the correlation between X + Y and Y + Z is:
If X ∼ N (0, 1) and Y = X2 then the correlation coefficient r (X, Y) is
Calculate the correlation coefficient between the following values :
x: 3, 5, 1, 7, 5
y: 4, 3, 0, 8, 2
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
