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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Estimat<strong>in</strong>g the sample mean<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 />
Suppose we take a r<strong>and</strong>om sample <strong>of</strong> women <strong>and</strong> measure their<br />
heights.<br />
Some assumptions:<br />
1 Each woman’s height is a normal r<strong>and</strong>om variable X i with<br />
mean µ <strong>and</strong> variance σ;<br />
2 The variables X i are <strong>in</strong>dependent.<br />
The strong law <strong>of</strong> large numbers says that with probability one,<br />
as n → ∞,<br />
∑ n<br />
i<br />
X i<br />
ˆµ = → µ<br />
n<br />
∑ n<br />
i X<br />
So ˆµ = i<br />
n<br />
is a good estimator for µ.
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