Determining the Constant k in a Continuous Random Variable’s PDF

Understanding the concept of the Probability Density Function (PDF) is fundamental in statistics, especially when dealing with continuous random variables. In this article, we will explore how to determine the constant k in a given PDF for a specific random variable. This is an essential step in ensuring that the function adheres to the properties of a valid probability distribution, particularly the requirement that the total probability over the entire range is equal to 1.

Introduction to Probability Density Functions (PDFs)

Before diving into the specific example, let's briefly recap the basics of a PDF. The probability density function is a function that describes the relative likelihood for a continuous random variable to take on a given value. A key property of a PDF is that the total area under the curve (from negative infinity to positive infinity) must be equal to 1. This property allows us to calculate probabilities for intervals of the random variable by integrating the PDF over that interval.

The Problem Presented

We are given a specific continuous random variable X with the following Probability Density Function (PDF) expressed as:

$$ f_X (x) frac{x}{5k}, quad 0 leq x leq 3 $$

The task is to determine the value of the constant k. This is a common exercise in probability theory, often seen in introductory statistics and calculus courses.

Deriving the Value of k

To solve for the constant k, we need to ensure that the integral of the given PDF over its defined domain equals 1. This is a fundamental requirement for the PDF to be valid. Mathematically, this can be expressed as:

$$ int_0^3 f_X (x) , dx 1 $$

Substituting the given PDF into the integral:

$$ int_0^3 frac{x}{5k} , dx 1 $$

Next, we can factor out the constant (frac{1}{5k}) from the integral:

$$ frac{1}{5k} int_0^3 x , dx 1 $$

The integral of (x) from 0 to 3 can be solved as follows:

$$ int_0^3 x , dx left[ frac{x^2}{2} right]_0^3 frac{9}{2} $$

Substituting this back into the equation:

$$ frac{1}{5k} cdot frac{9}{2} 1 $$

Solving for k, we get:

$$ frac{9}{10k} 1 $$

Which simplifies to:

$$ 10k 9 $$

Therefore, we find that:

$$ k frac{9}{10} cdot frac{1}{k} frac{9}{10} cdot frac{1}{k} frac{9}{10} cdot frac{1}{9/10} frac{9}{10} cdot frac{10}{9} frac{1}{10} $$

Thus, the value of the constant k that makes the given function a valid PDF is 30:

$$ k frac{1}{30} $$

Therefore, the complete PDF of the random variable (X) is:

$$ f_X (x) frac{x}{5 cdot frac{1}{30}}, quad 0 leq x leq 3 $$

$$ f_X (x) frac{3}{5}, quad 0 leq x leq 3 $$

$$ f_X (x) 6x, quad 0 leq x leq 3 $$

Conclusion

In conclusion, understanding and solving for the constant (k) in a given Probability Density Function (PDF) is a crucial step in ensuring that the function meets the necessary criteria for a valid probability distribution. This process involves setting up an integral that equals 1 and solving for (k). By following the steps outlined in this article, one can determine the correct value of (k) and ensure that the PDF accurately represents a continuous random variable over its given range.

Further Reading and Resources

To delve deeper into this topic, consider exploring the following resources:

Probability Density Function on Wikipedia Khan Academy: Probability Density Functions Wolfram MathWorld: Probability Density Function