Question

Let \( (X, Y) \) have the joint probability density function

\[ f_{X,Y}(x,y) = \begin{cases} \frac{4}{(x + y)^3}, & x>1, y>1 \\ 0, & \text{otherwise} \end{cases} \]

Then which one of the following statements is NOT true?

• A. The probability density function of \( X + Y \) is \[ f_{X+Y}(z) = \frac{4}{z^3}(z - 2), \, z>2, \, \text{otherwise}. \]
• B. \( P(X + Y>4) = \frac{3}{4} \)
• C. \( E(X + Y) = 4 \log 2 \)
• D. \( E(Y | X = 2) = 4 \)

Hint:

To find the expected value of a sum of random variables, use the probability density function of the sum and integrate it over the appropriate range.

Answer 1

The Correct Option is C

We are given the joint probability density function \( f_{X,Y}(x,y) \), and we need to evaluate the provided statements. 
Step 1: Find the probability density function of \( X + Y \). 
The given joint probability density function for \( X \) and \( Y \) can be used to find the probability density function of \( X + Y \), denoted by \( f_{X+Y}(z) \). The correct form of \( f_{X+Y}(z) \) is: \[ f_{X+Y}(z) = \frac{4}{z^3}(z - 2), \, z>2. \] This corresponds to option (A), so it is true. 
Step 2: Evaluate \( P(X + Y>4) \). 
To find \( P(X + Y>4) \), we integrate the probability density function \( f_{X+Y}(z) \) from 4 to infinity: \[ P(X + Y>4) = \int_4^\infty \frac{4}{z^3}(z - 2) \, dz = \frac{3}{4}. \] So, option (B) is also true. 
Step 3: Evaluate \( E(X + Y) \). 
The expected value of \( X + Y \) is given by: \[ E(X + Y) = \int_2^\infty z f_{X+Y}(z) \, dz. \] Using the correct form of the density function, the expected value \( E(X + Y) \) does not equal \( 4 \log 2 \), and this makes option (C) false. 
Final Answer: \[ \boxed{\text{(C) } E(X + Y) = 4 \log 2 \text{ is NOT true}}. \]