1' BERNOULLI DISTRIBUTION

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1. BERNOULLI DISTRIBUTION A trial with only two
possible outcomes is used so frequently as a
building block of a random experiment that it is
called a Bernoulli trial. It is usually assumed
that the trials that constitute the random
experiment are independent. This implies that the
outcome from one trial has no effect on the
outcome to be obtained from any other trial.
Furthermore, it is often reasonable to assume
that the probability of a success on each trial
is constant.
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• Consider the following random experiments and
  random variables. Do they meet the following
  criteria
• Does the experiment consist of Bernoulli trials?
  
• Are the trials that constitute the random
  experiment are independent?
• Is probability of a success on each trial is
  constant?
• 1. Flip a coin 10 times. Let X the number of
  heads obtained.
• 2. Of all bits transmitted through a digital
  transmission channel, 10 are received in error.
  Let X the number of bits in error
• a) in the next 4 bits transmitted.
• b) during next 4 hours

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Let us express the outcomes as a random variable
X, letting X 1 stand for S and X 0 stand for
F. ? ProbX 1, (1-?) ProbX 0 The
mean of the distribution E(X) The
variance Var(X)
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2. BINOMIAL DISTRIBUTION
Binomial Experiment 1) n identical (Bernoulli)
trials 2) Each trial has two outcomes success or
failure 3) Probability of success, ?, remains
constant from trial to trial 4) Trials are
independent
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Consider, for example, the arbitrary sequence of
nine trials, S, S, F, S, F, F, F,
S, S. The probability of this sequence, assuming
independence is ? ? (1-?) ? (1-?)
(1-?) (1-?) ? ? ?5(1-?)4 More generally,
we will say that the probability of getting a
sequence of n trials which contains r successes
and (n - r) failures is ?r (1 - ?)n - r
However, this is not the probability of the
event "r successes in n trials", since there are
several different ways for this event to occur.
How many?
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Binomial Random Variable
X number of successes in a binomial
experiment Then the probability of r successes
is given by
r 0, 1, 2 , , n
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Mean and Variance of a Binomial Random Variable

• Let X be a Binomial random variable with
  parameters n and ?

Mean ?x ? n ? Variance Standard
Deviation
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Lecture Exercise 1
Batches that consist of 50 coil springs from a
production process are checked for conformance to
customer requirements. The mean number of
nonconforming coil springs in a batch is 5.
Assume that the number of nonconforming springs
in a batch, denoted as X, is a binomial random
variable. (a) What are n and p? (b) What is P(X
1)? (0.0286) (c) What is P(X lt
1)? (0.0338) ( P(X0) 0.0052
) (d) What is P(X gt 49)?
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The Binomial distribution has many applications
in such problems as quality control. Given a
certain probability ? of a component failing,
(which maybe is regarded as the maximum
acceptable) we can compute the probability of
finding r failed components in a randomly chosen
batch of n components.
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3. GEOMETRIC DISTRIBUTION
X number of failures prior to the first
success in a sequence of Bernoulli
trials
r 0, 1, 2 , The first success is to come on
the (r 1) trial, and has to be preceded by r
failures
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Example 1
If the probability is 0.05 that a part produced
by a machine will be defective, what is the
probability that the 6th part produced on a given
day is the first one that is defective?
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4. POISSON DISTRIBUTION

• Consider e-mail messages that arrive at a mail
  server on a computer network.
• The number of events over an interval (such as
  the number of messages that arrive in one hour)
  is a discrete random variable that is often
  modeled by a Poisson distribution
• Poisson process
• Consider the number of times that an event occurs
  over a period of time or space, and assume that
• 1) The probability that the event occurs is the
  same for any two intervals of equal length. and,
• 2) The occurrences of the event are independent
  for any non-overlapping intervals, then.

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Poisson Random Variable
X the number of occurrences of an event in a
(Poisson) process Then the probability of x
occurrences is given by
? is the mean (expected) number of occurrences in
an interval, and e2.71828 is the base of the
natural logarithms
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• It is important to use consistent units in the
  calculation of probabilities, means, and
  variances involving Poisson random variables. The
  following example illustrates unit conversions.
  For example,
• average number of flaws per millimeter of wire is
  3.4
• average number of flaws in 10 millimeters of wire
  is 34
• average number of flaws in 100 millimeters of
  wire is 340.
• If a Poisson random variable represents the
  number of counts in some interval,
• then the mean of the random variable must equal
  the expected number of counts in the same length
  of interval.

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Mean and Variance of a Poisson Random Variable
Let X be Poisson random variable with parameter
?
Mean ?x ? ? Variance Standard
Deviation
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Example 2
Flows occur at random along the length of a thin
copper wire. Suppose that the number of flows
follows a Posson distribution with a mean of 2.3
flows per mm. a) Determine the probability of
exactly 2 flows in 1 mm of wire. b) 10 flows in
5mm of wire c) At least one flow in 2mm of wire.
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Example 2
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Exercise 2 - 1999 exam question
4. a) Messages arrive to a computer server
according to a Poisson distribution with a mean
rate of 10 per hour. What is the probability
that 3 messages arrive in 15 minutes?