Random variables

STAT1003 – Statistical Techniques

Dr. Emi Tanaka

Australian National University

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Table of contents

• Random variables
• Probability distribution
• Parametric distributions

Random variables

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon.

There are two main types of random variables:

• Discrete random variables: possible values are a countable number of distinct values
  • E.g., number of heads in 10 coin tosses.
• Continuous random variables: any values within a given range
  • E.g., the height of students in a class.

Notation

• Random variables are usually represented by uppercase letters (e.g., \(X\) or \(Y\)).
• Specific outcomes (realised values) are represented by corresponding lowercase letters (e.g., \(x\) or \(y\)).

• For example:
  • Let \(X\) be the number of heads in 3 coin tosses.
  • Then \(x = 0\) represents the outcome of getting 0 heads.

Probability distribution

A probability distribution describes how probabilities are assigned to the possible values of a random variable.

• For a discrete random variable, the distribution is described by a probability mass function (pmf).
• For a continuous random variable, the distribution is described by a probability density function (pdf).

By convention we denote:

• the pmf of a discrete random variable \(X\) as \(P(X = x)\) or \(p_X(x)\), and
• the pdf of a continuous random variable \(X\) as \(f_X(x)\).

• Probability distribution summarises characteristics of the population.
• Barplots for discrete data and histograms and density plots for continuous data are visualisations of estimates of the probability distribution of the underlying random variable.

Parametric distributions

• Parametric distribution are defined by just a handful of parameters.

Bernoulli distribution

viewof p2 = Inputs.range([0, 1], {step: 0.01, label: "p", value: 0.5})

Binomial distribution

viewof n = Inputs.number([1, Infinity], {step: 1, label: "n", value: 10})
viewof p = Inputs.range([0, 1], {step: 0.01, label: "p", value: 0.5})

Poisson distribution

viewof lambda = Inputs.number([0, Infinity], {step: 0.1, label: "λ", value: 1})

Negative binomial distribution

viewof nbsize = Inputs.number([1, Infinity], {step: 1, label: "n", value: 10})
viewof nbprob = Inputs.range([0, 1], {step: 0.01, label: "p", value: 0.5})

Normal distribution

viewof mu2 = Inputs.number({step: 0.5, label: "μ", value: 0})
viewof sd2 = Inputs.number({step: 0.01, label: "σ", value: 1})

t distribution

viewof df2 = Inputs.number([0, Infinity], {step: 1, label: "degrees of freedom", value: 1})

Uniform distribution

viewof xmin = Inputs.number({step: 0.1, label: "min", value: 0})
viewof xmax = Inputs.number({step: 0.1, label: "max", value: 1})

Gamma distribution

viewof shape = Inputs.number({step: 0.1, label: "shape", value: 1})
viewof rate = Inputs.number({step: 0.1, label: "rate", value: 1})