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
uniform 2

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.

continuous uniform dist pdf
PDF (Uniform distribution continuous)
uniform cdf
CDF (Uniform distribution continuous)

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.

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 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)

pdf exponential
PDF (exponential)
cdf exponential
CDF (exponential)

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.

Mean: αβ
Variance: αβ2

gamma pdf
PDF (Gamma distribution)
gamma cdf
CDF (Gamma distribution)

The Beta Distribution

X has a Beta distribution with parameters α > 0 and β > 0. denoted by X~Beta(α, β), if
beta distribution

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.

Mean:mean of beta distribution
Variance: variance

There are 3 more important distributions – Beta distribution, t and Cauchy distribution and χ2 distribution which we will discuss later.

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