# Continuous Random Variables

### Uniform Probability Distribution

X has Uniform(a, b) distribution, written X~Uniform(a, b), ifwhere 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, σ^{2}), 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.

### Exponential 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}

### Gamma Distribution

For α > 0, the Gamma function is defined by Γ(α) = ∫_{0}^{∞}y^{α-1}e^{-y}dy.

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 χ^{2} distribution which we will discuss later.