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)

The correct answer is -1.

Let us break down the solution step-by-step to understand how this result is obtained.

Step 1: Properties of the exponentially distributed random variables
We are given two exponentially distributed random variables X and Y, both having a mean (expected value) of 0.50.
For an exponentially distributed random variable with parameter λ, the mean is given by:

$$E(X)=\frac{1}{\lambda }$$
Given that the mean is 0.50, we have:

$$\frac{1}{{\lambda }_1}=\frac{1}{{\lambda }_2}=0.50$$
This gives the rate parameters: ${\lambda }_1={\lambda }_2=2$.

The variance of an exponentially distributed random variable is given by:

$$Var(X)=\frac{1}{{\lambda }^2}$$
Therefore, the variances of X and Y are:

$$Var(X)=\frac{1}{2^2}=0.25$$
$$Var(Y)=\frac{1}{2^2}=0.25$$

Step 2: Relate the variance of Z to the covariance of X and Y
We are defined a new random variable Z = X + Y. The variance of the sum of two random variables is expressed as:

$$Var(Z)=Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)$$
We are given that Var(Z) = 0. Substituting the values we have:

$$0=0.25+0.25+2Cov(X,Y)$$
$$0=0.50+2Cov(X,Y)$$
Solving for Cov(X, Y):

$$2Cov(X,Y)=-0.50$$
$$Cov(X,Y)=-0.25$$

Step 3: Calculate the correlation coefficient r
The correlation coefficient r between two random variables X and Y is given by the formula:

$$r=\frac{Cov(X,Y)}{\sqrt{Var(X)\cdot Var(Y)}}$$
Substituting the calculated values into the formula:

$$r=\frac{-0.25}{\sqrt{0.25\cdot 0.25}}=\frac{-0.25}{0.25}=-1$$
Thus, the value of the correlation coefficient r is indeed -1.