Transforming Probability Density Functions (PDFs) for Random Variables

Understanding the transformation of probability density functions (PDFs) is a crucial skill in probability and statistics. This article will guide you through the process of finding the PDF of a new random variable W 3 - X, given that X is a random variable with a density given by FX(x) 3/4 x^2 I[-1,1].

Step-by-Step Guide to PDF Transformation

Given:

The PDF of X, FX(x) 3/4 x^2 I[-1,1]. We need to find the PDF of W, where W 3 - X.

1. Finding the CDF of X

The cumulative distribution function (CDF) of X is given by:

FX(x) ∫_${-1}^x$ (3/4 t^2 dt)

Integrating the given PDF, we have:

FX(x) [1/4 t^3]_${-1}^x 1/4 x^3 - 1/4 (-1)^3 1/4 x^3 1/4

Thus:

FX(x) 1/4 (x^3 1)

Note that the CDF of a continuous random variable should range between 0 and 1. Here, when x 1, FX(x) 1/2, which is within the range. Therefore, the correct PDF is:

FX(x) 3/2 x^2 I[-1,1]

Thus, the corrected CDF is:

F_x(x) 1/2 (x^3 1), for -1 ≤ x ≤ 1

When x 1, F_x(x) 1, which is the correct behavior of the CDF at the upper bound.

2. Transforming the Random Variable W 3 - X

To find the PDF of W, we need to consider the transformation:

W 3 - X

Then, solving for X, we get:

X 3 - W

The support for W can be found by considering the range of X:

-1 ≤ 3 - W ≤ 1

Solving these inequalities, we get:

2 ≤ W ≤ 4

The new CDF of W, FW(w), is given by:

FW(w) 1 - F_x(3 - w)

Substituting the CDF of X:

FW(w) 1 - [1/2 ((3 - w)^3 1)]

Thus:

FW(w) 1 - 1/2 (27 - 27w 9w^2 - w^3 1) 1 - 14 13.5w - 4.5w^2 0.5w^3

Simplifying:

FW(w) -0.5 (28 - 27w 9w^2 - w^3)

Therefore, the PDF of W, FW(w), is obtained by differentiating the CDF:

FW(w) -0.5 (-27 18w - 3w^2)dw/dw

Simplifying:

FW(w) 0.5 (27 - 18w 3w^2)

Thus, the final PDF of W is:

FW(w) 0.5 (27 - 18w 3w^2) I[2,4]

Alternative Method: Using the Transformation Formula

Alternatively, you can use the transformation formula to find the PDF of W:

PDF of W PDF of X * |dX/dW|

Here, W 3 - X, so:

dX/dW -1

Thus, the PDF of W is:

FW(w) (3/2 (3 - w)^2) * |-1| (3/2 (3 - w)^2)

Therefore, the PDF of W is:

FW(w) 0.5 (27 - 18w 3w^2) I[2,4]

Conclusion

By following the steps above, we have successfully transformed the PDF of the random variable X to the PDF of the new random variable W. This process involves finding the CDF, applying the transformation, and differentiating the new CDF to obtain the final PDF. The support for the new random variable W, 2 ≤ W ≤ 4, has also been determined.

Additional Resources

For further understanding and detailed practice, you may want to explore the following resources:

Probability and Statistics textbook focusing on random variable transformations. Online tutorials and courses on probability theory and statistical methods. Practice problems and solutions available on platforms like Khan Academy, Coursera, and Udemy.

Understanding the transformation of PDFs is essential for a deeper grasp of statistical methods and their applications in various fields such as finance, engineering, and data science.