The derivative of the probability density function (PDF), P(x), is the rate at which the probability density changes with respect to the variable x. Based on the provided reference, this derivative is equivalent to the second derivative of the cumulative distribution function (CDF), D(x).
Understanding the Relationship Between CDF and PDF
The reference defines the probability density function (PDF) P(x) for a continuous probability distribution through its relationship with the cumulative distribution function (CDF) D(x). The core relationship is that the PDF is the derivative of the CDF.
According to the reference:
- The CDF is defined as D(x) = P(X ≤ x) (equation 4), representing the probability that the random variable X is less than or equal to x.
- The CDF can also be expressed as the integral of the PDF: D(x) = ∫_(-∞)^x P(ξ) dξ (equation 5).
The reference explicitly states the derivative relationship:
- D'(x) = P(x) (as shown in equation 3), meaning the derivative of the CDF at point x gives the value of the PDF at that point.
Finding the Derivative of the PDF
The question asks for the derivative of the PDF, P(x). Using the relationship established in the reference, P(x) = D'(x), we can find its derivative by applying standard calculus:
If P(x) = D'(x)
Taking the derivative of both sides with respect to x:
d/dx [P(x)] = d/dx [D'(x)]
This yields:
P'(x) = D''(x)
Therefore, the derivative of the probability density function P(x) is the second derivative of the cumulative distribution function D(x).
Summary of Key Relationships
| Function | Notation | Relationship | Derivative |
|---|---|---|---|
| Cumulative Distribution | D(x) | D(x) = ∫_(-∞)^x P(ξ) dξ | D'(x) = P(x) (from reference eq. 3 & 5) |
| Probability Density | P(x) | P(x) = D'(x) (from reference eq. 3) | P'(x) = d/dx [P(x)] = D''(x) |
| Derivative of PDF | P'(x) | P'(x) = D''(x) | P''(x) = d/dx [P'(x)] = D'''(x) |
Practical Context
While the reference focuses on the definition, the derivative of the PDF, P'(x), provides insights into the shape of the probability distribution.
- P'(x) > 0: The probability density is increasing at x.
- P'(x) < 0: The probability density is decreasing at x.
- P'(x) = 0: The PDF has a critical point at x (potentially a peak, valley, or inflection point). Locating where P'(x) = 0 is useful for finding modes (peaks) of the distribution.
In essence, the derivative of the PDF tells us about the instantaneous rate of change of the density function itself.
