Lecture 7. A) Random Sample

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Random Sample

Let [math]X=\left(X_{1}..X_{n}\right)[/math] be an n-dimensional random vector. The random variables [math]X_{1}..X_{n}[/math] constitute a random sample if they are (mutually) independent and have identical (marginal) distributions.

We usually refer to such variables as being i.i.d.: Independent and Identically distributed. To reiterate, these variables share the same distribution, and are not correlated.

It follows that if [math]X[/math] is a random sample from distribution [math]F\left(\cdot\right)[/math], then

[math]F_{X_{1}..X_{n}}\left(x_{1}..x_{n}\right)\underset{(independence)}{\underbrace{=}}\Pi_{i=1}^{n}F_{X_{i}}\left(x_{i}\right)\underset{(F_{X_{i}}=F,\,\forall i)}{\underbrace{=}}\Pi_{i=1}^{n}F\left(x_{i}\right)[/math].

Also, note that the multiplicative result also applies to the pmd and pdf.