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).Correct answer is '-1'. Can you explain this answer?

Question

Answer

Given Information:
- X and Y are exponentially distributed random variables with a mean of 0.50.
- Z = X * Y
- Variance of Z is 0.

Solution:

To find the correlation coefficient (r) between X and Y, we first need to find the covariance between X and Y.

Covariance:
The covariance between two random variables X and Y is given by the formula:

Cov(X, Y) = E[(X - E[X])(Y - E[Y])]

Since X and Y are exponentially distributed random variables with a mean of 0.50, their expected values are both 0.50.

E[X] = 0.50
E[Y] = 0.50

So, the covariance between X and Y becomes:

Cov(X, Y) = E[(X - 0.50)(Y - 0.50)]

Since the variance of Z is 0, we can write:

Var(Z) = E[Z^2] - E[Z]^2

Since Var(Z) = 0, we have:

E[Z^2] - E[Z]^2 = 0

Variance of Z:
The variance of Z can be written as:

Var(Z) = Var(X*Y)

Using the properties of variance, we can expand it as:

Var(Z) = Var(X) * Var(Y) + Var(X) * [E[Y]]^2 + [E[X]]^2 * Var(Y)

Since X and Y are exponentially distributed with a mean of 0.50, their variances are both (0.50)^2 = 0.25.

Var(X) = Var(Y) = 0.25

Substituting these values, we get:

Var(Z) = 0.25 * 0.25 + 0.25 * (0.50)^2 + (0.50)^2 * 0.25
= 0.0625 + 0.0625 + 0.0625
= 0.1875

But given that Var(Z) = 0, this implies:

0.1875 = 0

This is a contradiction.

Conclusion:
Since the equation Var(Z) = 0 leads to a contradiction, the assumption that the variance of Z is 0 is not possible. Therefore, the value of r cannot be determined.

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