Probability for Electrical and Computer Engineering

1
Probability for Electrical and Computer
Engineering

• EE387
• Chintha Tellambura
• Lecture 7

2
Today Outline

• The Binomial Probability Law
• Multinomial Probability Law
• The Geometric Probability Law
• Pages 62-66 in ALG

3
Goals

• Upon completion of today lecture, you should be
  able to
• Understand differences between the
  probability laws
• Know the conditions under which these apply
• Compute probabilities under these laws.

4
The Binomial Probability Law

• A Bernoulli trial involves performing an
  experiment once and noting whether a particular
  event A occurs (success) or does not occur
  (failure).
• To find the probability of k successes in n
  independent Bernoulli trials, the outcome of a
  single Bernoulli trial can be viewed as the
  outcome of a toss of a coin.

5
Theorem Binomial Probability Law

• If k is the number of successes in n independent
  Bernoulli trials, the probabilities of k are
  given by the binomial probability law
• where is the binomial coefficient
• and p is the probability of a success in a
  singleBernoulli trial.

6
Binomial theorem

• If a b 1, we get
• There are 2n distinct possible sequences of
  successes and failures in n trials.

7
Binomial theorem

• Naturally
• where
• For large values of n the following recursive
  formula can be used to determine pn(k)

8
Multinomial Probability Law

• It is a generalization of the binomial
  probability law to the case of non-binary
  partition.
• Let B1,B2, . . . ,BM be a partition of the sample
  space S.
• and
• Suppose n independent repetitions of the
• experiment are performed and kj be the number
• of times Bj occurs in n trials. Thus,
• (k1, k2, . . . , kM) are the number of times
• B1,B2, . . . ,BM occur.

9
Multinomial Probability Law

• The probability of (k1, k2, . . . , kM) is given
  by the multinomial probability law
• where
• The binomial probability law is a special case
  for M 2 (binary partition).

10
The Geometric Probability Law

• We repeat independent Bernoulli trials until the
  occurrence of the first success.
• Let m be the number of trials carried out until
  the occurrence of the first success Probability
  that m trials are required to achieve success is
• where Ai - "success in i-th trial",
• i 1, 2, . . . ,m.

11
Geometric Probability

• We observe the following
• The probability that more than k trials
  arerequired to achieve success

12
Summary

• We have discussed
• The Binomial Probability law
• Multinomial Probability Law
• The Geometric Probability Law