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How to Find the Median of a Probability Density Function?

Published in Probability Distributions 3 mins read

The median of a probability density function (PDF) is the value that divides the distribution into two equal halves. This means the area under the curve to the left of the median is 0.5, and the area to the right is also 0.5. Here's how to find it:

Steps to Determine the Median

  1. Understand the Definition: The median, often denoted as m, is the value such that:

    P(X ≤ m) = 0.5

    where X is the random variable.

  2. Set Up the Integral: For a probability density function f(x), you need to solve the following equation for m:

    ∫-∞m f(x) dx = 0.5

    In other words, the integral of the PDF from negative infinity (or the lower bound of the distribution if it's not -∞) up to the median m must equal 0.5. Alternatively, you can also use:

    ∫m∞ f(x) dx = 0.5

    This calculates the integral from the median m up to positive infinity (or the upper bound of the distribution).

  3. Solve for m: This is the crucial step. Evaluate the integral and solve the resulting equation for m. This may involve algebraic manipulation or numerical methods, depending on the complexity of the PDF.

Example

Let's say we have a PDF given by f(x) = (3/2)x2 for -1 ≤ x ≤ 1, and 0 otherwise. To find the median:

  1. Set up the integral:

    ∫-1m (3/2)x2 dx = 0.5

  2. Evaluate the integral:

    [(1/2)x3]-1m = 0.5

    (1/2)*m3 - (1/2)(-1)3 = 0.5

    (1/2)*m3 + 1/2 = 0.5

  3. Solve for m:

    (1/2)*m3 = 0

    m3 = 0

    m = 0

Therefore, the median of this distribution is 0.

Key Considerations

  • PDF Definition: Ensure you have the correct and complete definition of the PDF, including its domain (the range of x-values for which the PDF is valid).
  • Integration Techniques: Be proficient in integration, as you'll need to evaluate definite integrals.
  • Equation Solving: Depending on the PDF, solving for m might require advanced algebraic or numerical techniques. If the integral is difficult to solve analytically, numerical integration methods can be used to approximate the value of m.

In summary, finding the median of a probability density function involves setting up and solving an integral equation where the integral of the PDF from the lower bound of its domain to the median equals 0.5. This process requires a good understanding of calculus and potentially numerical methods for more complex PDFs.

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