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probability

# probability - 0.1 Introduction Here are some of the typical...

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0.1 Introduction Here are some of the typical uses of randomness in parallel computation. The first use is for coordinating processors; usually random methods are faster than deterministic ones, or in someways inferior. For example, if two processors contend for the same resource, then we can flip a coin to decide who gets it. A more general application is what is called “symmetry breaking” – there are some P processors wanting one of some k resources. The problem is how to coordi- nate who gets what, and so that as many processors as possible are given a resource. One idea would be to each processor make a request to a random resource: this would distribute the processors reasonably evenly amongst the resources, and then use coin tossing to remove further contention. In this case, deterministic methods might require more computation, or might not distribute the load uniformly. A related use is in packet routing algorithms, where we use random priorities for messages. This effectively puts messages into batches, with each batch getting roughly the same number on the average. Notice that this is a strong statement: by randomly assigning messages into batches we are likely to get a good partitioning on all links in the system. Whereas it is very easy for deterministic methods to get a perfectly good partitioning on a single link, but doing it simultaneously on all links is quite hard. 1 Probability Probability space is defined as the set of all the possible outcomes of an experiment. This is also known as sample space . Let us denote the sample space by P . Any subset of the probability space is known as an event . Example 1: If we perform the experiment of flipping two coins then the probability space is C 2 = { tt, th, ht, hh } . The event that there is atleast one tail is given by { tt, th, ht } . Example 2: If the experiment performed is drawing a card from the deck, then the probability space will be the entire deck ( D ). A probability distribution is a function P from the set of events to < . This function should satisfy the following axioms of probability. Axioms of Probability 1. For any event A , P ( A ) 0 2. P ( P ) = 1, 3. For any infinite sequence of disjoint events A 1 , A 2 , . . . , P [ i =1 A i ! = X i =1 P ( A i ) 1.1 Basic Properties of Probability 1. P ( ) = 0. 1

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2. For any finite sequence of disjoint events A 1 , A 2 , . . . A n P n [ i =1 A i ! = n X i =1 P ( A i ) 3. For any event A , P ( A c ) = 1 - P ( A ). 4. For any event A , 0 P ( A ) 1. 5. If A B , then P ( A ) P ( B ). 1.2 Random Variables A random variable X is a function from the probability space to the set of real numbers. Formally, X : P → < . Example 1: Let H 2 be the no. of heads in 2 coin flips, hence in our sample space C 2 , we have, H 2 ( tt ) = 0 , H 2 ( ht ) = 1 , H 2 ( th ) = 1 , H 2 ( hh ) = 2. Example 2: Let C be 1 if the card drawn is a club, 0 otherwise. Let A be 1 if the card drawn is an ace, 0 otherwise.
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probability - 0.1 Introduction Here are some of the typical...

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