Central Limit Theorem

1
Central Limit Theorem

• Let X1, X2, , Xn be n independent, identically
  distributed random variables with mean m and
  standard deviation s. For large n
• Sn X1X2Xn is approximately normal with mean
  nm and standard deviation .
• The average of the random variables (i.e., the
  sample mean) is approximately normal with mean m
  and standard deviation .

2
(No Transcript)
3
(No Transcript)
4
0.3
0.5
0.2
Suppose at each time step a particle has
probability 0.3 of moving 1 step to the left,
probability 0.5 of moving 1 step to the right and
probability 0.2 of staying where it is. Find the
probability that after 10,000 time steps the
particle is no more than 1000 steps to the right
of its starting point.
5
Conditional Expectation Given an Event

• The conditional expectation of a random variable
  Y given an event A, denoted E(YA), is the
  expectation of Y under the conditional
  probability distribution given A

6
Rule of Average Conditional Expectations

• For any random variable Y with finite expectation
  and any discrete random variable X,
• Another way of writing the above is