Continuous Distributions Updated (1)

Continuous Random Variables

  • Continuous random variables differ from discrete variables in that they can take any value over an interval, such as (a,b), (-∞, 0), (0,∞), and (-∞,∞).

  • The probability distribution associated with a continuous random variable X is known as the probability density function (pdf), denoted by f(x).

  • Events are defined as the area under the density curve for specific ranges of X, making cumulative distribution functions (cdf) meaningful: P(X ≤ x).

Key Concepts

Probability Density Function (PDF) and Cumulative Density Function (CDF)

Applications of Continuous Random Variables

  • Commonly modeled variables in business include:

    • Weights of products (cans, packages).

    • Delivery and completion times (projects, packages).

    • Financial metrics (returns, costs).

    • Rates (hotel occupancy, churn, etc.).

Continuous Probability Distributions

Characteristics of Density Curves

  • Each density curve indicates the likelihood of all possible outcomes, where the total area under the curve equals 1.

  • For any single value of X, the probability is zero: P(X = c) = 0.

    • Probabilities are discussed for intervals (regions).

Probability Density Function Requirements

  1. f(x) ≥ 0 for every x in [a,b].

  2. The total area under the curve from a to b equals 1.

  3. To find the probability of a range (c ≤ X ≤ d), compute the area under the graph of f(x) from c to d.

Uniform Distribution

  • A continuous random variable X follows a uniform distribution, denoted X ~ U(a,b), if it has a constant pdf over the interval.

  • Formula:

Example - Uniform Distribution

  • For X ~ Uniform(2, 6), pdf is:

    • f(x) = 0.25 for 2 ≤ x ≤ 6.

  • Probability that the bus is late between 3 and 5 minutes:

    • P(3 ≤ X ≤ 5) = (5 - 3) / (6 - 2) = 0.5.

Normal Distribution

Overview

  • The normal distribution is a key continuous distribution characterized by its bell shape.

    • Represents many real-world phenomena.

    • Defined by mean (μ) and variance (σ²).

  • Notable properties:

    • Symmetric, with mean = median = mode.

    • Range: (-∞, +∞).

    • Approximately 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.

Transformation to Standard Normal Distribution (Z)

  • Any normally distributed variable can be standardized to Z:

    • Z = (X - μ) / σ.

    • The standard normal distribution has a mean of 0 and standard deviation of 1.

Finding Normal Probabilities

  • To calculate the probability P(a < X < b), Standardize X to Z-value and use the Z-table to find probabilities:

    • Example:

      • For a normally distributed X with μ = 18 and σ = 5, to find P(X < 18.6):

        • Z = (18.6 - 18) / 5 = 0.12.

  • Use cumulative distribution tables or statistical software for calculations.

Empirical Rule

  • The Empirical Rule approximates the dispersion of data in a normal distribution:

    • About 68% data falls in the range μ ± 1σ.

    • About 95% data falls in the range μ ± 2σ.

    • About 99.7% data falls in the range μ ± 3σ.

Exponential Distribution

Definition

  • A right-skewed distribution used to model time between events (e.g., waiting times).

  • Key parameters include λ (lambda), which represents the rate of occurrence.

Key Properties

  • The pdf for an exponential variable:

    • f(x) = λe^{-λx}, for x ≥ 0.

  • Mean and standard deviation are both equal to 1/λ.

Example

  • If customers arrive at a rate of 15 per hour, the average inter-arrival time is:

    • P(T < 3 minutes) = 1 - e^{-λx} for λ = 15/60 and x = 3/60.

Binomial to Normal Approximation

  • As the number of trials n increases, Binomial distributions can be approximated as Normal if np ≥ 10 and n(1 - p) ≥ 10.

  • Example Calculations:

    • For n=100 and p=0.5, μ = np = 50, σ = sqrt(np(1-p)) = 5.

These notes provide a comprehensive overview of continuous random variables, including key distributions like uniform, normal, and exponential, their properties, and applications in statistical modeling.