Final answer:
To find the CDF and PDF of |xy|, we need to consider the joint distribution of X and Y as independent random variables uniformly distributed in the interval 0, 1. The CDF and PDF of |xy| are derived using integration and differentiation methods. The CDF depends on the value of z, while the PDF is 0 except for the interval 0 < z ≤ 1.
Step-by-step explanation:
To find the Cumulative Distribution Function (CDF) and Probability Density Function (PDF) of |xy|, we need to consider the joint distribution of X and Y as they are independent random variables uniformly distributed in the interval 0, 1.
Let Z = |X * Y|. We will find the CDF and PDF of Z.
CDF of Z
To find the CDF of Z, we need to integrate the joint PDF over the region where |xy| ≤ z.
Case 1: If z ≤ 0, the probability is 0, since the absolute value of a positive number is always positive.
Case 2: If 0 < z ≤ 1, the probability can be calculated as follows:
P(|xy| ≤ z) = P(X ≤ z/Y) + P(-X ≤ z/Y)
Since X and Y are independent, we can rewrite the above expression as:
P(|xy| ≤ z) = P(X ≤ z) * P(1/Y ≤ z) + P(-X ≤ z) * P(1/Y ≤ z)
As X and Y are uniformly distributed in the interval 0, 1, their individual CDFs are given by:
F(X) = X, F(Y) = Y
Substituting these values into the expression for P(|xy| ≤ z):
P(|xy| ≤ z) = z * (1/z) + z * (1/z) = 2
Therefore, the CDF of |xy| is:
F(Z) = {0, z ≤ 0; 2, 0 < z ≤ 1; 1, z > 1}
PDF of Z
To find the PDF of Z, we differentiate the CDF with respect to z:
f(Z) = dF(Z)/dz
Since F(Z) is a piecewise function, the PDF can be written as:
f(Z) = {0, z ≤ 0; 0, 0 < z ≤ 1; 0, z > 1}