pdf, pmf, cdf <-> derivative and integral
The relationship between the probability density function (PDF), probability mass function (PMF), and cumulative distribution function (CDF) can indeed be understood in terms of derivatives and integrals, similar to the relationship between these mathematical operations. Here’s a breakdown:
1. Concepts:
PDF (Probability Density Function):
- Used for continuous random variables.
- Represents the likelihood of a random variable taking on a specific value.
- The area under the PDF curve over an interval gives the probability that the random variable falls within that interval.
PMF (Probability Mass Function):
- Used for discrete random variables.
- Represents the probability of a random variable taking on a specific value directly.
CDF (Cumulative Distribution Function):
- Used for both discrete and continuous random variables.
- Represents the probability that a random variable is less than or equal to a certain value.
2. Relationships:
For Continuous Random Variables:
The PDF is the derivative of the CDF:
\begin{equation}f_X(x) = \frac{d}{dx} F_X(x),\end{equation} where \(F_X(x)\) is the CDF, and \(f_X(x)\) is the PDF.The CDF is the integral of the PDF: \begin{equation}F_X(x) = \int_{-\infty}^x f_X(t) \, dt\end{equation}
For Discrete Random Variables:
The PMF is analogous to the derivative (in a discrete sense): \begin{equation}P(X = x_i) = F_X(x_i) - F_X(x_{i-1}),\end{equation} where \(x_i\) is a discrete value of the random variable, and \(F_X(x_i)\) is the CDF.
The CDF is the summation (analogous to an integral) of the PMF: \begin{equation}F_X(x_k) = \sum_{x_i \leq x_k} P(X = x_i)\end{equation}
3. Comparison with Derivatives and Integrals:
| Concept | Derivative/Integral Analogy | Explanation |
|---|---|---|
| PDF or PMF | Derivative | Represents the rate of change (density or mass at a point). |
| CDF | Integral (or summation for PMF) | Accumulates probability up to a certain point. |
| PMF to CDF (Discrete) | Summation | Like the integral for continuous, sums probabilities. |
| PDF to CDF (Continuous) | Integral | Integrates density over a range to get probability. |
4. Practical Insight:
- The PDF/PMF is more detailed, focusing on specific points or small intervals of the random variable.
- The CDF gives a broader cumulative view, showing probabilities up to a certain threshold.
- differentiation (continuous case) or differences (discrete case) and integration (continuous case) or summation (discrete case).