Overview of basic concepts in Statistics and Probability - SAMSI
Overview of basic concepts in Statistics and Probability - SAMSI
Overview of basic concepts in Statistics and Probability - SAMSI
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R<strong>and</strong>om variables<br />
<strong>Overview</strong> <strong>of</strong><br />
<strong>basic</strong> <strong>concepts</strong><br />
<strong>in</strong> <strong>Statistics</strong><br />
<strong>and</strong><br />
<strong>Probability</strong><br />
Avanti<br />
Athreya<br />
Prelim<strong>in</strong>aries<br />
Important<br />
distributions,<br />
scal<strong>in</strong>g laws,<br />
<strong>and</strong> the CLT<br />
Parametric<br />
estimation <strong>and</strong><br />
hypothesis<br />
test<strong>in</strong>g<br />
A r<strong>and</strong>om variable X is a function from the sample space Ω<br />
<strong>in</strong>to Euclidean space R n ; X : Ω → R n .<br />
The doma<strong>in</strong> <strong>of</strong> a r<strong>and</strong>om variable is the SAMPLE SPACE.<br />
Hence X is only “r<strong>and</strong>om” because its doma<strong>in</strong> consists <strong>of</strong><br />
experimental outcomes that cannot be determ<strong>in</strong>ed <strong>in</strong> advance.<br />
Once the outcome ω ∈ Ω is known, however, X(ω) is not<br />
r<strong>and</strong>om.<br />
Example: Let X=number <strong>of</strong> heads <strong>in</strong> two tosses <strong>of</strong> a co<strong>in</strong>.<br />
1 The sample space Ω = {HH,HT,TH,TT }.<br />
2 Note that X(HH) = 2, X(HT) = 1, X(TH) = 1,<br />
X(TT) = 0.
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