Continuous Random Variables

Uniform Probability Distribution

X has Uniform(a, b) distribution, written X~Uniform(a, b), if
uniform
where a < b. The distribution function is
uniform 2

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.

Exponential Distribution

X has an Exponential Distribution with parameter β > 0, denoted by X ~ Exp(β), if
exponential

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) = ∫0yα-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.

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