Chapter3 Discrete Random Variables

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Chapter-3 Discrete Random Variables
Probability Distribution (Cont.)
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Discrete Uniform Distribution

• Simplest random variable assumes only a finite
  number of possible values, each with equal
  probability.
• A random variable X has a discrete uniform
  distribution if each of the n values in its
  range, say, x1, x2, , xn, has equal probability.
  Then,
• f(xi) 1/n

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Example 3-13
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Discrete Uniform Distribution (Cont.)

• Suppose X is a discrete uniform random variable
  on the consecutive integers a, a1, a2, , b,
  for a lt b. The mean of X is
• The variance of X is

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Discrete Uniform Distribution (Cont.)

• Example 3-14.
• If all the values in the range of a random
  variable X are multiplied by a constant (without
  changing any probabilities), the mean and
  standard deviation of X are multiplied by the
  constant. The variance should be multiplied by
  the constant squared.
• Example 3-15.

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REVISION
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Binomial distribution

• Examples
• Flip a coin 10 times. X Number of heads
  obtained.
• Worn machine tool produces 1 defective parts. X
  Number of defective parts in the next 25 parts
  selected.
• In the next births at a hospital. X Number of
  female births.
• Multiple choice test contains 10 questions, each
  with four choices, and you guess at each
  question. X Number of questions answered
  correctly.

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Binomial distribution (Cont.)

• Bernoulli Trial A trial with only two possible
  outcomes.
• 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.

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Binomial distribution (Cont.)

• The number of ways of partitioning n objects into
  two groups, one of which is of size x, is
• For Success and Failure trials, the above
  expression equals the total number of different
  sequences of trials that contain x successes and
  n-x failures.
• Remember n! n(n-1)(n-2)(2)(1), 1! 1, 0! 1

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• Example 3-16 (refer to example 3-4 Figure 3.1)

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Binomial distribution (Cont.)

• A random experiment consists of n Bernoulli
  trials such that
• (1) The trials are independent.
• (2) Each trial results in only two possible
  outcomes, labeled as success and failure.
• (3) The possibility of a success in each trial,
  denoted as p, remains constant.

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Binomial distribution (Cont.)

• The random variable X that equals the number of
  trials that result in a success has a binomial
  random variable with parameters 0 lt p lt 1 and n
  1, 2,
• The probability mass function of X is
• (X 0, 1, 2, , n)

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Binomial distribution (Cont.)

• For a fixed n, the distribution becomes more
  symmetric as p increases from 0 to 0.5 or
  decreases from 1 to 0.5.
• For a fixed p, the distribution becomes more
  symmetric as n increases.
• Example 3-18.
• If X is a binomial random variable with
  parameters p and n,
• ? E(X) np ?2 V(X) np(1-p)
• Exercises 3-61, 3-66.

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REVISION

• Bernoulli distribution
• (1) The trials are independent.
• (2) Each trial results in only two possible
  outcomes, labeled as success and failure.
• (3) The possibility of a success in each trial,
  denoted as p, remains constant.
• ? E(X) np
• ?2 V(X) np(1-p)

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Poisson Distribution

• Examples
• (1) Particles of contamination in semiconductor
  manufacturing.
• (2) Flaws in rolls of textiles.
• (3) Calls to a telephone exchange.
• (4) Power outages.
• (5) Atomic particles emitted from a specimen.

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Poisson Distribution

• Given an interval of real numbers, assume counts
  occur at random throughout the interval. If the
  interval can be partitioned into subintervals of
  small enough length such that
• (1) The probability of more than one count in a
  subinterval is zero,
• (2) The probability of one count in a subinterval
  is the same for all subintervals and proportional
  to the length of the subinterval,
• (3) The count in each subinterval is independent
  of other subintervals, THEN
• The random experiment is called a Poisson
  process.

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Poisson Distribution (Cont.)

• The random variable X that equals the number of
  counts in the interval is a Poisson Random
  Variable with parameters 0 lt ? , and the
  probability mass function of X is
• The distribution can be applied to time interval,
  area interval, etc.

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Poisson distributions for selected values of the
parameters
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Poisson Distribution (Cont.)

• If the Poisson random variable represents the
  number of counts in some interval, the mean of
  the random variable must equal the expected
  number of counts in the same length of interval.
• It is important to use consistent units in the
  calculation of probabilities, means, and
  variances involving Poisson random variables.
• Example
• If average number of flaws per millimeter of
  wire is 3.4, then the average number of flaws in
  10 millimeters of wire is 34.
• Examples 3-32, 3-33

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Poisson Distribution (Cont.)

• If X is a Poisson random variable with parameter
  , then
• ? E(X) ? ?2 V(X) ?
• THUS, the mean and variance of a Poisson random
  variable are equal.
• If the variance of count data for the
  distribution is much greater than the mean of the
  same data, the Poisson distribution is not a good
  model for the distribution of the random
  variable.

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Probability models

• Binomial distribution X denotes the number of
  trials that result in a success out of n trials.
• Geometric distribution X denotes the number of
  trials until the first success.
• Negative binomial distribution X denotes the
  number of trials required to obtain r successes.
• Poisson distribution X denotes the number of
  counts in some interval.

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ANNOUNCEMENTS

• Assignment III
• 4, 5, 15, 17, 27, 28, 36, 38, 39, 45, 47, 48
• Assignment IV
• 56, 57, 62, 64, 68, 98, 100, 101, 102