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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 = X1+X2+…+Xn is approximately normal with
mean nm and standard deviation n s .
– The average of the random variables (i.e., the
sample mean) is approximately normal with mean
m and standard deviation s n .
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.
Conditional Expectation Given
an Event
• The conditional expectation of a random
variable Y given an event A, denoted
E(Y|A), is the expectation of Y under the
conditional probability distribution given A:
E Y A   y PY  y | A
all y
Rule of Average Conditional
Expectations
• For any random variable Y with finite
expectation and any discrete random
variable X,
E Y    E Y | X  x P X  x 
all x
• Another way of writing the above is
EY   EEY | X 