This figure below describes the joint PDF of the random variables X and Y. These random variables take values in [0,2] and [0,1], respectively. At x<=1, the value of the joint PDF is 1/2. At 2>x>1 the value of the joint PDF is 3/2. What's the marginal PDF of x if 1<x<2?
To find the marginal PDF of X when 1 < X < 2, we need to integrate the joint PDF with respect to Y over the range of [0,1].
Since the joint PDF is given as follows:
1/2 for X <= 1 and 0 <= Y <= 1
3/2 for 1 < X < 2 and 0 <= Y <= 1
For 1 < X < 2, the marginal PDF of X is given by integrating the joint PDF over the range of Y from 0 to 1:
marginal PDF of X = ∫(joint PDF) dy (from 0 to 1)
= ∫(3/2) dy (from 0 to 1)
= (3/2) * (1 - 0)
= 3/2
Therefore, the marginal PDF of X when 1 < X < 2 is 3/2.
It has to be in terms of x
Apologies for the confusion.
To find the marginal PDF of X when 1 < X < 2, we need to integrate the joint PDF with respect to Y over the range of [0,1].
Since the joint PDF is given as follows:
1/2 for X <= 1 and 0 <= Y <= 1
3/2 for 1 < X < 2 and 0 <= Y <= 1
For 1 < X < 2, the marginal PDF of X is given by integrating the joint PDF over the range of Y from 0 to 1:
marginal PDF of X = ∫(joint PDF) dy (from 0 to 1)
= ∫(3/2) dy (from 0 to 1)
= (3/2) * (1 - 0)
= 3/2
Therefore, the marginal PDF of X when 1 < X < 2 is 3/2.
To find the marginal PDF of X if 1 < X < 2, we need to integrate the joint PDF with respect to Y over the range [0,1].
The joint PDF is given as follows:
For X ≤ 1, f(x, y) = 1/2
For 2 > X > 1, f(x, y) = 3/2
Since we are interested in the marginal PDF of X, we need to find the probability density function for X.
First, let's express the joint PDF as a piecewise function:
f(x, y) =
1/2 if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
3/2 if 1 < x < 2, 0 ≤ y ≤ 1,
0 otherwise.
Since we are only interested in the range 1 < X < 2, we are concerned with the second part of the joint PDF equation:
f(x, y) = 3/2 if 1 < x < 2, 0 ≤ y ≤ 1.
To find the marginal PDF of X, we integrate this equation with respect to y over the range [0,1]:
∫[0,1] (3/2) dy
Performing the integration, we get:
3/2 * [0,1]
= 3/2.
Therefore, the marginal PDF of X if 1 < X < 2 is 3/2.