8.
Random experiment
“A random experiment is an experiment,
trial, or observation that can be repeated
numerous times under the same conditions...
It must in no way be affected by any previous
outcome and cannot be predicted with
certainty.” (http://cnx.org/content/m13470/latest/)
i.e. it is uncertain (we don’t know ahead of
time what the answer will be) and repeatable
(ideally).
Monday, 22 March 2010
9.
Where we are
Univariate random variables:
an experiment with one output
Bivariate random variables:
an experiment with two outputs
Sequences of random variables:
An experiment performed repeatedly.
Repeatable = i.i.d
Monday, 22 March 2010
10.
A sampling distribution:
Summary statistics from
a repeated experiment
Monday, 22 March 2010
11.
Deﬁnitions
Sample = results of n random experiments.
Random sample = result of a random
experimented repeated n times. Therefore,
they’re iid.
Both are sequences of random variables.
Statistic = A function of random variables
with no unknown parameters.
Monday, 22 March 2010
12.
Example
Spin a bottle and record the angle in
degrees in which it points. Repeat.
How would you write this
mathematically?
Monday, 22 March 2010
19.
What will happen as I
8000
vary the number of
samples I average
over? (What theorem
6000 applies here?)
count
4000
2000
0
100 150 200 250
V1
Monday, 22 March 2010
26.
Theory
We’ll start with the mean of normally
distributed random variables, then try to
extend in various ways.
Monday, 22 March 2010
27.
Your turn
X1, X2, ... are iid N(μ, σ2)
n
¯ Sn
Sn = Xi Xn =
n
1
Find their mgfs. What do you notice?
Hint: MX (t) = exp µt + σ t 2 2
Monday, 22 March 2010
28.
Reading
4.2, 4.2.1
4.2.2, 4.4
Monday, 22 March 2010
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