Handouts:

Course description

Description of the course, grading, policy on late HWs.

The web page is www-inst.eecs.berkeley.edu/~cs174 and contains lecture summaries and an active course syllabus.

Course Content, three parts:

An example problem. The birthday problem (with a twist for added privacy)

From the table below, copy the number under the month of your birthday onto a piece of paper.

Now from one of the tables below, copy the number under your birth date onto the same piece of paper.

Now add the two numbers. If the answer is bigger than 372, subtract 372 from it. You should now have a number between 1 and 372.

The next step is a survey of how many people came up with numbers in the following ranges

The goal is to find a group with 8 or more people in it. Such a group will almost certainly have two people with the same number. Those people share a common birthday. People in this group will say their number, until we catch two with the same number. The rest of us will know that they share a birthday, but not what the birthday is.

We can even protect the identities of the individuals with the same birthday, even though they themselves can find out who the person with the same birthday is. How could we do that? Hint: everyone in class would have to read out their number.

This example illustrates several points:

Probability implies that its very likely certain things happen (two people in class have the same birthday).

We can use probability analysis to design a good algorithm for the problem (finding the people).

This algorithm is usually fast, even though a "worst case" algorithm would be slow.

We can use encryption to hide information from onlookers.

We can use randomization to selectively share information.

Normally, variables are symbols that hold values from some domain, like the integers. A random variable is a variable which has a probability distribution on its domain. e.g.

X Î {1,…,6} is a variable, and if we specify in addition

Pr[X=i] = 1/6 for i Î 1,…,6 then X is a random variable. The probability distribution in this case is the uniform distribution.

We can also have an exponential distribution on an infinite domain Pr[X=i] = 1/2ⁱ for i = 1, 2, 3,… but of course the sum

Independence

A very important concept for this course is independence of RV's. X₁ and X₂ are independent random variables if Pr[X₁=u, X₂=v] = Pr[X₁=u] Pr[X₂=v] for all u and v in the domains.

Associated with a random variable is its expected value E[X], defined by

E[X] = å i Pr[X=i]

Linearity

Another important idea for this course: Expected value satisfies E[X₁ + X₂] = E[X₁] + E[X₂] (linearity). Note: Linearity of expectation doesnt require independence.

Proof:

E[X₁ + X₂] = å_i   å_j   (i + j) Pr[X₁=i, X₂=j]

= å_i å_j i Pr[X₁=i, X₂=j]  +  å_i å_j j Pr[X₁=i, X₂=j]

= å_i i å_j   Pr[X₁=i, X₂=j]  +  å_j j å_i  Pr[X₁=i, X₂=j]    move constants outside the sums.

= å_i i  Pr[X₁=i]  +  å_j j Pr[X₂=j] collapse sums over all possible values of the other random variable.

= E[X₁] + E[X₂]     QED and nowhere did we use the independence property

Products

The rule for products of RV's is what you might expect. However, it does require independence.

Theorem: If X₁ and X₂ are independent, then E[X₁ X₂] = E[X₁] E[X₂]