3.2 Continuous probability distributions

Continuous probability distributions describe random variables that can take any value within a range. They're characterized by probability density functions (PDFs) and cumulative distribution functions (CDFs), which help calculate probabilities for specific intervals.

Common continuous distributions include normal, exponential, and uniform. These are used in various applications, from modeling natural phenomena to simulating random processes. Understanding their properties is crucial for statistical analysis and decision-making in management.

Properties and Characteristics of Continuous Probability Distributions

Properties of continuous probability distributions

• Continuous random variables take on any value within a given range represented by real numbers (height, weight)
• Probability density function (PDF) describes likelihood of different values occurring with area under curve representing probability
• Cumulative distribution function (CDF) gives probability of value being less than or equal to specific point
• Infinite number of possible values within range
• Probability of any single point is zero
• Total area under PDF curve equals 1
• Non-negative function throughout entire range

Probability density functions in distributions

• PDF mathematically describes relative likelihood of continuous random variable
• PDFs are non-negative for all values and integral over entire range equals 1
• Height of PDF curve indicates relative likelihood but not direct probability measure
• Probability found by integrating PDF over interval $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$
• Graphically represented as smooth curve for continuous distributions
• Forms basis for calculating probabilities used in hypothesis testing and parameter estimation

Applications and Calculations

Applications of common continuous distributions

• Normal distribution
  • Bell-shaped curve with parameters mean ($\mu$) and standard deviation ($\sigma$)
  • Standard normal distribution uses z-score
  • Applied to natural phenomena (height, weight, test scores)
• Exponential distribution
  • Models time between events with rate parameter ($\lambda$)
  • Exhibits memoryless property
  • Used for random processes (customer arrivals, equipment failures)
• Uniform distribution
  • Constant probability density over interval with minimum (a) and maximum (b) values
  • Applied in simulations (random number generation, rounding errors)

Cumulative distribution function for probabilities

• CDF defined as $F(x) = P(X \leq x)$
• CDF is integral of PDF: $F(x) = \int_{-\infty}^{x} f(t) dt$
• CDF properties:
  • Monotonically increasing
  • Limits: $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$
• Probability calculations: $P(a < X \leq b) = F(b) - F(a)$
• Inverse CDF (Quantile function) finds specific percentiles
• Applied in statistical inference (confidence intervals, hypothesis testing)