Continuous Distributions Overview

• Continuous distributions involve random variables that can take infinitely many values.

• Key components:

  • Probability Density Function (pdf)

  • Cumulative Distribution Function (cdf)

Random Variables Types

• Discrete Random Variable: Takes distinct, separate values (e.g., counts).

• Continuous Random Variable: Takes any value in a range on the real line (e.g., measurements like height, waiting time).

Probability Density Function (pdf)

• Defines the positive probability of outcomes such that the total area under the pdf curve equals 1.

• The probability that a continuous random variable lies between two values is given by the area under the pdf curve between those values.

Important Concepts

• P(X = x) for continuous variables yields zero because of infinite outcomes.

• Probabilities described by inequalities are meaningful:

  • It is useful to discuss probabilities such as P(a < X < b) rather than specific values.

Cumulative Distribution Function (cdf)

• The cdf for a continuous random variable summarizes probabilities for ranges (e.g., P(X < x)).

• The relationship between pdf and cdf:

  • P(a < X < b) = cdf(b) - cdf(a).

Key Continuous Distributions

• Continuous Uniform Distribution: All values in a given range are equally likely;

  • Example: If X ~ Unif(a, b), then pdf is rectangular.

• Normal Distribution: Symmetric around the mean; characterized by mean (µ) and standard deviation (σ).

  • Standard Normal Distribution is a normal distribution with µ = 0 and σ = 1.

  • Allow for Z-scores to facilitate comparison across different scales.

Probability Calculations

• Use the pdf and cdf to calculate probabilities for continuous random variables.

• For example, for a random variable X ~ Normal(µ,σ): use Z-scores to find probabilities from the standard normal distribution table.