Question 1192049: Given f(x, y) = e^(-x-y), for x>0 and y>0
(a) Find the marginal density of X
(b) Find the conditional density of Y given X = x
(c) Find the first joint moment of X and Y
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Certainly, let's break down the 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!
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