2.2 Probability Mass and Density Functions

Probability Mass Functions (PMFs) and Probability Density Functions (PDFs) are key tools for describing random variables. PMFs assign probabilities to specific values for discrete variables, while PDFs represent probability density for continuous variables.

Understanding PMFs and PDFs is crucial for calculating probabilities and expected values. They have important properties like non-negativity and normalization. Cumulative Distribution Functions (CDFs) are related concepts that provide a different way to represent probability distributions.

Probability Mass and Density Functions

Definition of PMFs and PDFs

• Probability Mass Function (PMF) assigns probabilities to specific values for discrete random variables (dice rolls, number of customers)
• Probability Density Function (PDF) represents probability density over a range of values for continuous random variables (height, weight)
• PMF denoted as $P(X = x)$ or $f_X(x)$, PDF denoted as $f_X(x)$
• PMF gives exact probabilities while PDF gives probability densities
• Area under PDF curve represents probability, not the function value itself

Interpretation of PMFs and PDFs

• Discrete random variables using PMF
  • Calculate probability of specific value $P(X = x)$
  • Find probability of range by summing individual probabilities
• Continuous random variables using PDF
  • Probability of specific value always zero
  • Calculate probability of range by integrating PDF over interval $P(a \leq X \leq b) = \int_a^b f_X(x) dx$
• Expected value calculation differs for discrete and continuous variables
  • Discrete: $E[X] = \sum_x x \cdot P(X = x)$
  • Continuous: $E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) dx$

Properties of valid PMFs and PDFs

• Non-negativity ensures probabilities or densities are never negative
  • PMF: $P(X = x) \geq 0$ for all $x$
  • PDF: $f_X(x) \geq 0$ for all $x$
• Total probability (normalization) sums to 1 for complete probability space
  • PMF: $\sum_x P(X = x) = 1$
  • PDF: $\int_{-\infty}^{\infty} f_X(x) dx = 1$
• Support of function defines range of values where function is non-zero
• Cumulative distribution function exhibits monotonicity
• PDFs often possess continuity and differentiability properties

Calculation of CDFs

• Cumulative Distribution Function (CDF) represents probability of random variable being less than or equal to a value
• CDF denoted as $F_X(x) = P(X \leq x)$
• Discrete random variables CDF calculated by summing PMF values
  • $F_X(x) = \sum_{t \leq x} P(X = t)$
• Continuous random variables CDF found by integrating PDF
  • $F_X(x) = \int_{-\infty}^x f_X(t) dt$
• CDF properties include
  • Non-decreasing function
  • Right-continuous for discrete variables
  • Continuous for continuous variables
  • Approaches 0 as x approaches negative infinity
  • Approaches 1 as x approaches positive infinity