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].
Mean: 1/2 * (a+ b)
Variance: 1/12 * (b – a)2
We will learn to derive this later in the expectation section.
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 the Central Limit theorem. Suppose we draw a sample of observation of a random variable from a distribution independent of each other. The average of those samples of observation converges to normal distribution.
Normal Distribution is widely used in Statistics. So we need to study intricate details of this distribution. To know more about this distribution, see this.
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 components and the waiting times between rare events. Sometime you might also see the alternate parameterized form as,
ƒ(x) = λe-λβ
Mean: β (1/λ)
Variance: β2 (1/λ2)
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.
The Beta Distribution
X has a Beta distribution with parameters α > 0 and β > 0. denoted by X~Beta(α, β), if
In Beta distribution, there are two terms :
Γ(α + β) / Γ(α)Γ(β) . . . (i) and
xα-1(1 – x)β – 1 . . . (ii)
The first term (i) is constant and is used to ensure that the total probability is 1.
There are 3 more important distributions – Beta distribution, t and Cauchy distribution and χ2 distribution which we will discuss later.