let x and y be continuous random variables having joint density f(xy)=a^2 e^-ay, 0<=X<=Y.
Find the marginal densities of X an Y.
To find the marginal densities of X and Y, we need to integrate the joint density function over the respective variables.
Let's start with finding the marginal density of X. We integrate the joint density over the entire range of Y while keeping X fixed:
f_X(x) = ∫(from 0 to ∞) f(x, y) dy
Substituting the given joint density function:
f_X(x) = ∫(from 0 to ∞) a^2 e^(-ay) dy
To evaluate this integral, we can pull out the constant term a^2 and use the integration formula for e^(ax):
f_X(x) = a^2 * ∫(from 0 to ∞) e^(-ay) dy
= a^2 * [(-1/a) * e^(-ay)] (from 0 to ∞)
= a^2 * [(-1/a) * e^(-a∞) - (-1/a) * e^(-a*0)]
= a^2 * [0 - (-1/a)]
= a
Therefore, the marginal density of X is f_X(x) = a.
Now, let's find the marginal density of Y. We integrate the joint density over the entire range of X while keeping Y fixed:
f_Y(y) = ∫(from 0 to y) f(x, y) dx
Substituting the given joint density function:
f_Y(y) = ∫(from 0 to y) a^2 e^(-ay) dx
Again, we can pull out the constant term a^2 and integrate:
f_Y(y) = a^2 * ∫(from 0 to y) e^(-ay) dx
= a^2 * [(-1/a) * e^(-ay)] (from 0 to y)
= a^2 * [(-1/a) * e^(-ay) - (-1/a) * e^(-a*0)]
= a^2 * [(-1/a) * e^(-ay) + 1/a]
= (1/a) * (1 - e^(-ay))
Therefore, the marginal density of Y is f_Y(y) = (1/a) * (1 - e^(-ay)).
In summary:
The marginal density of X is f_X(x) = a.
The marginal density of Y is f_Y(y) = (1/a) * (1 - e^(-ay)).