Discrete Random Variables and Probability Distributions

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• Discrete Random Variables and Probability
  Distributions

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Discrete Random Variable

• Example
• A voice communication system for a business
  contains 48 external lines. At particular time,
  the system is observed, and some of lines are
  being used.
• Let X denote the number of line are being used.
• Then X can assume any of integer values 0-48.
• If 10 lines are in used, x 10.

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Probability Distributions and Probability Mass
Functions

• The probability distribution of a random
  variable X is a description of the probability
  associated with the possible values of X.
• Example There is a chance that a bit transmitted
  through a digital transmission channel is
  received in error.
• Let X of bits in error in the next four bits.
• Then X (0, 1, 2, 3, 4). Suppose that the
  probability are
• P(X 0) 0.6561, P(X 1) 0.2916, P(X 2)
  0.0486
• P(X 3) 0.0036, P(X 4) 0.0001

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Probability Distributions and Probability Mass
Functions
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Probability Distributions and Probability Mass
Functions

• For a discrete random variable X with possible
  values x1, x2,, xn a probability mass function
  is a function such that

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Cumulative Distribution Functions

• Example From previous example, we interest in
  the probability of three or fewer bits being in
  error.
• Then P(X 3) P(X 0) P(X 1) P(X 2)
  P(X 3)
• 0.65610.29160.04860.0
  036 0.9999
• or P(X 3) P(X 3) - P(X 2) 0.0036

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Cumulative Distribution Functions

• The cumulative distribution function of a
  discrete random variable X, denoted as F(x), is
• For a discrete random variable X, F(x) satisfies
  the following

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Mean and Variance

• Mean is a measure of the center or middle of the
  
• probability distribution.
• Variance is a measure of the dispersion, or
• variability in the distribution.
• Two different distributions can have the same
  mean
• and variance.

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Mean and Variance

• The mean or expected value of the discrete
  random variable X, denoted as µ or E(X), is
• The variance of X, denoted as s2 or V(X), is
• The standard deviation of X is

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Mean and Variance

• Example There is a chance that a bit transmitted
  through a digital transmission channel is
  received in error.
• Let X of bits in error in the next four bits.
• Then X (0, 1, 2, 3, 4). Suppose that the
  probability are
• P(X 0) 0.6561, P(X 1) 0.2916, P(X 2)
  0.0486
• P(X 3) 0.0036, P(X 4) 0.0001

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Mean and Variance

• V(X) s2 0.36
• s 0.6

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Mean and Variance

• Example The number of messages sent per hour
  over a computer network as the following.

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Discrete Uniform Distribution

• 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,
• Suppose X is a discrete uniform random variable
  on the consecutive integers a, a1, a2,, b,
  for a b.

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Discrete Uniform Distribution

• Example ?????????????????????????????????????????
  48 ??????
• ??????????????????????????????????????????????????
  ? ??????????????????
• ?????????????????????????????
• X (0, 1, 2, .., 48)

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Binomial Distribution

• Each trial has result either a success or a
  failure.
• The trial with two possible outcome is used as a
  building block of a random experiment called
  Bernoulli trial.
• Trials are independent implied that outcome from
  one trial has no effect on the outcome to be
  obtained from any other trial.
• Probability of success in each trial, denoted as
  p, is constant.

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Binomial Distribution

• The number of outcomes that contain x errors is
  needed.
• The random variable X that equals the number of
  trial that result in success with parameters 0 lt
  p lt1 and n 1, 2,. The probability mass
  function of X is

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Binomial Distribution

• Mean
• Variance

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Binomial Distribution

• Example ?????????????????????????????????????????
  ???????????
• ?????????????????? 0.1 ???????????????????????????
  ???????????????
• ???????????????
• Let X of bits in errors ????????????????? 4
  ?????????????
• ??? E bit in error
• O bit is okay
• ?????????????????????????????????????????????????
  2 bits

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Binomial Distribution
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Binomial Distribution

• ???????????? X 2 ??????
• (EEOO, EOEO, EOOE, OEEO, OEOE, OOEE)
• ??????? P(EEOO) P(E) P(E) P(O) P(O) (0.1)2
  (0.9)2 0.0081
• P(X 2) 6(0.0081) 0.0486
• ??????????

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Binomial Distribution

• Example ?????????????????????????? 10
  ??????????????????????????? ??????????????????????
  ??????????????????????? ??????????????????????????
  ???????? 18 ???????? ?????? 2 ????????????????????
  ?????????????????????
• Let X ??????????????????????????????????????????
  ??
• ??????? X ??? Binomial random variable with p
  0.1, n 18

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Binomial Distribution

• ?????????????????????????? 4 ?????????????????????
  ????????????????????
• ???????????????????????? 3 ???????????????????????
  ?? 7

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Geometric Distribution

• Closely related to the binomial distribution.
• Independent trials with constant probability p of
  a success on each trial.
• Instead of a fixed number of trials, trials are
  conducted until a success is obtained.
• Let the random variable X denote the number of
  trials until the first success.

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Geometric Distribution

• Then X is a geometric random variable with 0 lt p
  lt1 and
• Mean
• Variance

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Geometric Distribution

• Example ???????????????? wafer
  ??????????????????????????????? 0.01
• ??????????????????????????????????
  ???????????????????? wafer ????? 125
• ??????????????????????????????????????????????????
  ????????
• Let X ???????????????????????????????????????
  ???????????????????
• ??????? X ??? Geometric random variable with p
  0.01

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Geometric Distribution
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Geometric Distribution

• Defined as the number of trials until the first
  success.
• The count of the number of trials until the next
  success can be started at any trial without
  changing the probability distribution of the
  random variable.
• The probability of an error remains constant for
  all transmissions that is said to lack of memory
  property.

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Negative Binomial Distribution

• A generalization of a geometric distribution in
  which the random variable is the number of
  Bernoulli trials required to obtain r success
  results in the negative binomial distribution.
• Independent trial with constant probability (p)
  of success.
• Let X denote the number of trials until r success
  occur

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Negative Binomial Distribution

• Then X is a negative binomial random variable
  with 0 lt p lt 1 and r 1, 2, 3,, and
• Mean
• Variance

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Negative Binomial Distribution

• Example ?????????????????????????????????????????
  ????????????????????????????? 0.1
  ??????????????????????????????????????????????????
  ??????? ????????????????? 10 ?????
  ??????????????????????????????????????????????????
  ????? 4
• X ????????????????????????????????????????????
  ?????? 4
• ??????? X Negative binomial distribution (r
  4)
• ??????????????????? 3 ??????????????? 9 ????????
  ??????????? 10 ????
• ??????????????????????????????????? 4

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Negative Binomial Distribution

• ?????????? 9 ???????? ???? Binomial
  ???????????????? 3 ?????
• ???????? 10 ??????????????????????????????????????
  ? 4

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Hypergeometric Distribution

• Trials are not independent.
• There is a constant probability of a
  nonconforming part on each trial.
• The number of nonconforming parts in the sample
  is binomial random variable.

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Hypergeometric Distribution

• A set of N objects contains
• K objects classified as successes
• N-K objects classified as failures
• A sample of size n objects is selected randomly
  (without replacement) from the N objects, where K
  N and n N.

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Hypergeometric Distribution

• Let X denote the number of success in the sample
• Mean
• Variance

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Hypergeometric Distribution

• Example ????????????????? 850 ???????????????????
  ?????? 50 ???? ??????????????????????? 2
  ?????????????????????

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Hypergeometric Distribution

• Example ?????????? 300 ??????????????????????????
  ??????????????
• 200 ???? ???????????????????????? 100
  ??????????????????? 4 ??????????
• ????????????????????????????????????????????????
  ???????
• X ???????????????????
• ??????? X Hypergeometric distribution

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Hypergeometric Distribution

• ?????????????????????????????????????????????????
  2 ????
• ?????????????????????????????????????????????????
  1 ????

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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 the
  subinterval is the same for all subintervals and
  proportional to the length of the subinterval

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Poisson Distribution

• 3.The count in each subinterval is independent of
  other subintervals,
• the random experiment is called a Poisson
  process
• The random variable X that equals the number of
  counts in the interval is a Poisson with 0 lt ?
  and probability mass function of X

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Poisson Distribution

• Mean
• Variance

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Poisson Distribution

• Example ????????????????????????????????????????
  (Flaw)
• ??? ?????????????? Poisson ???????????? 2.3 ???
  ??? ????????? ????
• ????????????????????????? 2 ????? 1 ?????????
• X ????????????????????? 1 ?????????
• ????????????????????????? 10 ????? 5 ?????????

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Poisson Distribution

• ?????????????????????????????????? 1 ????? 2
  ?????????
• X ????????????????????? 1 ?????????
• E(X) 2 mm 2.3 flaws/mm 4.6 flaws