Question

Given Probability Density Function (PDF):( f_{XY}(x,y) = begin{cases} e⁻ˣ⁻ʸ & for x, y > 0 & otherwise end{cases}

What is the marginal probability density function of X?
a) e⁻ˣ
b) e⁻²ˣ
c) e⁻²ˣ⁻ʸ
d) 2e⁻ˣ⁻ʸ

Answer 1

The marginal probability density function of X is found by integrating the joint PDF concerning y over y's range, resulting in e^{-x} as the correct marginal PDF of X.

Step-by-step explanation:

The marginal probability density function of X can be found by integrating the joint probability density function concerning y. In this case, the joint probability density function is given by f_{XY}(x,y) = e⁻ˣ⁻ʸ for x, y > 0. To find the marginal probability density function of X, we integrate f_{XY}(x,y) concerning y from 0 to infinity: f_X(x) = ∫0∞f_{XY}(x,y) dy = ∫0∞e⁻ˣ⁻ʸ dy. Integrating e⁻ˣ⁻ʸ concerning y gives us -e⁻ˣ⁻ʸ evaluated from 0 to infinity. Substituting the limits of integration: f_X(x) = -e⁻ˣ⁻ʸ∣0∞ = -e⁻ˣ ∙ e⁻ˣ⁻ʸ e⁻⁰ = -e⁻ˣ (0 - 1) = e⁻ˣ. Therefore, the marginal probability density function of X is e⁻ˣ. Hence, the correct answer is (a) e⁻ˣ.