Distribution of a sample mean

STAT1003 – Statistical Techniques

Dr. Emi Tanaka

Australian National University

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Table of contents

• Distribution of a sample mean
• Case 1: \(N(\mu, \sigma^2)\)
• Case 2: \(U(a, b)\)
• Case 3: \(B(m, p)\)
• Case 4: \(\text{Poisson}(\lambda)\)
• Sampling distribution of the sample mean

Distribution of a sample mean

• Suppose \(X_1, X_2, \ldots, X_n\) are i.i.d. with mean \(\mu\) and variance \(\sigma^2\).
• Then \(\text{E}(\bar{X}) = \mu\) and \(\text{Var}(\bar{X}) = \dfrac{\sigma^2}{n}\).
  

• Let’s assume that the population is:

  1. \(N(\mu, \sigma^2)\)
  2. \(U(a, b)\) so \(\mu = \dfrac{a + b}{2}\) and \(\sigma^2 = \dfrac{(b - a)^2}{12}\).
  3. \(B(m, p)\) so \(\mu = m p\) and \(\sigma^2 = m p (1 - p)\).
  4. \(\text{Poisson}(\lambda)\) so \(\mu = \lambda\) and \(\sigma^2 = \lambda\).

• We then examine the empirical distribution of the sample mean.

Case 1: \(N(\mu, \sigma^2)\)

viewof mu = Inputs.range([-5, 5], {step: 1, label: "mu"})
viewof sigma = Inputs.range([1, 5], {step: 1, label: "sigma"})

Case 2: \(U(a, b)\)

viewof a = Inputs.range([-5, 0], {step: 1, label: "a"})
viewof b = Inputs.range([a, 5], {step: 1, value: 5, label: "b"})

Case 3: \(B(m, p)\)

viewof m = Inputs.range([1, 100], {value: 20, step: 1, label: "m"})
viewof p = Inputs.range([0, 1], {step: 0.01, label: "p"})

Case 4: \(\text{Poisson}(\lambda)\)

viewof lambda = Inputs.range([1, 20], {step: 1, label: "lambda"})

Sampling distribution of the sample mean

viewof n = Inputs.range([1, 100], {value: 20, step: 1, label: "n"})

What do you notice about the sampling distributions of the sample means?