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Derive a CDF from a PDF

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MediumData ScientistTechnical ScreenStatistics & Math

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This question tests the ability to derive a Cumulative Distribution Function (CDF) from a given Probability Density Function (PDF), ensuring proper integration and boundary conditions.

Problem Statement

Suppose a continuous random variable X represents the time (in hours) a rider waits for a ride-share vehicle. The probability density function (PDF) of X is given by:

f(x) = { k * x for 0 <= x < 2, k * (4 - x) for 2 <= x <= 4, 0 otherwise }

  1. Determine the value of the constant 'k' that makes f(x) a valid probability density function.
  2. Derive the cumulative distribution function (CDF), F(x), for all real numbers x.
  3. Verify your CDF by checking the boundary conditions and ensuring continuity at the point x = 2.

Constraints

  • The total probability must sum to 1.
  • The CDF must be defined for all x in the range (-infinity, +infinity).

Example

If f(x) were a uniform distribution on [0, 1], f(x) = 1, then F(x) = 0 for x 1.

Hints

Hint 1

Recall that the integral of the PDF over its entire support must equal 1, which will allow you to solve for the constant k.

Hint 2

The CDF F(x) is defined as the integral of the PDF from negative infinity to x; remember to calculate this piece-wise for each interval of the PDF.

Hint 3

Ensure that your final F(x) function is continuous at x = 0, x = 2, and x = 4, and that it maps the support to the range [0, 1].

Follow-up questions

  • How would the CDF change if the wait time X was conditioned on the event that the rider waited at least 1 hour?
  • If you were asked to estimate the median wait time from this distribution, how would you approach it using the CDF you just derived?
  • How does the property of the CDF being non-decreasing relate to the non-negativity constraint of the PDF?