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. #m/def/prob Since each value of 𝑥 is on its own massless¹, 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

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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