Continuous Random Variables
Uniform Probability Distribution
X has Uniform(a, b) distribution, written X~Uniform(a, b), if
where a < b. The distribution function is
The CDF can be calculated by integrating ƒ(x) from a to x for the interval [a, b].
Normal (Gaussian) Distribution
X has a Normal distribution, X ~ N(μ, σ2), if
Here μ is the center (or mean) and σ is the spread (or standard deviation) of the distribution. We say that X has a Standard Normal Distribution(Z) if μ =0 and σ = 1.
Normal distribution plays a vital role in statistics because of Central Limit theorem. Suppose we draw sample of observation of a random variable from a distribution independent of each other. The average of those sample of observation converges to normal distribution.
X has an Exponential Distribution with parameter β > 0, denoted by X ~ Exp(β), if
The exponential distribution is used to model the lifetimes of electronic component and the waiting times between rare events. Sometime you might also see the alternate parameterized form as,
ƒ(x) = λe-λβ
For α > 0, the Gamma function is defined by Γ(α) = ∫0∞yα-1e-ydy.
X has Gamma distribution with parameters α > 0 and β > 0, X ~ Gamma(α, β) if
The exponential distribution is just a Gamma(1, β) distribution.
There are 3 more important distributions – Beta distribution, t and Cauchy distribution and χ2 distribution which we will discuss later.