Central Limit Theorem

• Let X1, X2, …, Xn be n independent, identically distributed random variables with mean and standard deviation . For large n:– Sn = X1+X2+…+Xn is approximately normal with

mean n and standard deviation .– The average of the random variables (i.e., the

sample mean) is approximately normal with mean and standard deviation .

n

n

0.2

0.3 0.5

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:

y

AyYPyAYE all

|

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

x

xXPxXYEYE all

|

XYEEYE |