Question 1192146
Certainly, let's break down the given problem step-by-step.

**Given:**

* Joint probability density function: f(x, y) = e^(-x-y) for x > 0 and y > 0 

**a) Find the marginal density of X:**

* To find the marginal density of X, we integrate the joint density function with respect to y over its entire range:

   f_X(x) = ∫[from 0 to ∞] f(x, y) dy 
          = ∫[from 0 to ∞] e^(-x-y) dy
          = e^(-x) ∫[from 0 to ∞] e^(-y) dy
          = e^(-x) * [-e^(-y)] [from 0 to ∞]
          = e^(-x) * (0 - (-1)) 
          = e^(-x) 

   Therefore, the marginal density of X is:
   f_X(x) = e^(-x) for x > 0 

**b) Find the conditional density of Y given X = x:**

* The conditional density of Y given X = x is defined as:

   f_Y|X(y|x) = f(x, y) / f_X(x)

* Substituting the given values:

   f_Y|X(y|x) = e^(-x-y) / e^(-x) 
               = e^(-y) 

   Therefore, the conditional density of Y given X = x is:
   f_Y|X(y|x) = e^(-y) for y > 0

**c) Find the first joint moment of X and Y:**

* The first joint moment of X and Y, E[XY], is given by:

   E[XY] = ∫∫[over the support of (X, Y)] x * y * f(x, y) dx dy

* In this case, the support of (X, Y) is x > 0 and y > 0:

   E[XY] = ∫[from 0 to ∞] ∫[from 0 to ∞] x * y * e^(-x-y) dy dx

* We can solve this double integral using integration by parts. However, a quicker approach is to recognize that the joint density function is the product of the marginal densities of X and Y:

   f(x, y) = e^(-x-y) = e^(-x) * e^(-y) = f_X(x) * f_Y(y)

* This indicates that X and Y are independent random variables. 

* For independent random variables, E[XY] = E[X] * E[Y]

* We know that X and Y both follow an exponential distribution with parameter 1. The expected value of an exponential distribution with parameter λ is 1/λ.

* Therefore, 
   E[X] = 1/1 = 1 
   E[Y] = 1/1 = 1

* Hence, the first joint moment of X and Y is:
   E[XY] = E[X] * E[Y] = 1 * 1 = 1

**In summary:**

* Marginal density of X: f_X(x) = e^(-x) for x > 0
* Conditional density of Y given X = x: f_Y|X(y|x) = e^(-y) for y > 0
* First joint moment of X and Y: E[XY] = 1

I hope this comprehensive explanation is helpful!