Real random variable

Continuous random variable

A continuous random variable is a Real random variable with a cumulative distribution function that is differentiable everywhere, except maybe at the boundary. prob Since each value of 𝑥 is on its own massless^[See probability mass function.], probabilities are considered instead along intervals. The Probability density function 𝑓𝑋 :ℝ →ℝ is the derivative of the cumulative distribution function, whence by the fundamental theorem of calculus

ℙ(𝑎≤𝑋≤𝑏)=∫𝑏𝑎𝑓𝑋(𝑥)𝑑𝑥=𝐹𝑋(𝑏)−𝐹𝑋(𝑎)

Similarly we may define the Cumulative distribution function, which allows the equivalence of distributions.

Distributions

The most common distributions of continuous random variables are

• Uniform distribution
• Exponential distribution
• Normal distribution
• Logistic distribution
• Rayleigh distribution
• Log-normal distribution
• Beta distribution
• Gamma distribution (Chi-squared distribution)

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