Computing Fundamentals 2 Lecture 7 Statistics

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Computing Fundamentals 2Lecture 7 Statistics

• Lecturer Patrick Browne
• http//www.comp.dit.ie/pbrowne/
• Room K408

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Statistics

• Raw data are just lists of facts and numbers. The
  branch of mathematics that organizes, analyzes
  and interprets raw data is called statistics.

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Permutations Combinations

• P(n,r) n! / (n-r)!
• Permutations a, b, and c taken 2 at a time is
  32/16 ltsequencegt
• ltabgt,ltbagt,ltacgt,ltcagt,ltbcgt,ltcbgt
• C(n,r) n! /r! (n-r)!
• Combinations of a, b, and c taken 2 at a time is
  32/213ab,ac,bc set
• ab is the same combination as ba, but they are
  distinct permutations.

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Probability Calculations

• Conditional probability
• P(AE) P(A ? E)/P(E)
• Test for independence
• P(A ? B) P(A)P(B)
• Calculation of union
• P(A ? B) P(A) P(B) P(A ? B)

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Frequency Table

• One way of organizing raw data is to use a
  frequency table (or frequency distribution),
  which shows the number of times that an
  individual item occurs or the number of items
  that fall within a given range or interval.

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Frequency Distribution

• Suppose that a sample consists of the heights of
  100 male students at XYZ University. We arrange
  the data into classes or categories and determine
  the number of individuals belonging to each
  class, called the class frequency. The resulting
  table is called a frequency distribution or
  frequency table

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Frequency Distribution

• The first class or category, for example,
  consists of heights from 60 to 62 inches,
  indicated by 6062, which is called class
  interval. Since 5 students have heights belonging
  to this class, the corresponding class frequency
  is 5. Since a height that is recorded as 60
  inches is actually between 59.5 and 60.5 inches
  while one recorded as 62 inches is actually
  between 61.5 and 62.5 inches, we could just as
  well have recorded the class interval as 59.5
  62.5. In the class interval 59.5 62.5, the
  numbers 59.5 and 62.5 are often called class
  boundaries.

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Frequency Distribution

• The midpoint of the class interval, which can be
  taken as representative of the class, is called
  the class mark. A graph for the frequency
  distribution can be supplied by a histogram.

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Frequency table class interval
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Mean

• The arithmetic mean is the sum of the values in a
  data set divided by the number of elements in
  that data set.

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Mean

• The arithmetic mean is the sum of the values in a
  data set divided by the number of elements in
  that data set.
• x ?xi
• n
• x ?fixi where f
  denotes frequency
• ?fi

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Variance Standard Deviation

• List A 12,10,9,9,10
• List B 7,10,14,11,8
• The mean (x) of A B is 10, but the values of A
  are more closely clustered around the mean than
  those in B (or there is greater desperation or
  spread in B). We use the standard deviation to
  measure this spread (SD(A)1.1,SD(B) 2.4)

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Variance Standard Deviation

• The variance is always positive and is zero only
  when all values are equal.
• variance ?(xi - x )2
• n
• standard deviation

Alternatively
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Variance of a frequency distribution
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Median

• The median is the middle value. If the elements
  are sorted the median is
• Median valueAt(n1)/2 odd
• Median average(valueAtn/2,
• valueAtn/21) even
• For odd and even n respectively.
• Example 1,2,3,4,5 , Median 3
• Example 1,2,3,4,5,6, Median 3.5

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Mode

• The mode is the class or class value which occurs
  most frequently. We can have bimodal or
  multimodal collections of data.

The height of the bars is the number of cases in
the category
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Bernouilli Trials

• Independent repeated trial with two outcomes are
  called Bernouilli Trials. The probability of k
  successes in a binomial experiment is

• Where n is the number of trials and (n-k) is the
  number of failure.

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Bernouilli Trials Example

• John hits target p1/4,
• John fires 6 times, n6,
• What is the probability John hits the target 2
  times?

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Bernoulli Trials Example

• John hits target p1/4,
• John fires 6 times, n6,
• What is the probability John hits the target at
  least once?

No success (0), all failures, Anything to the
power of 0 is 1 Only 1 way to pick 0 from 6
Probability that John hits target at least once
EXCEL 1-((3/4)6)
Probability that John does not hit target
0 to the power 0 is undefined, anything else to
the power of zero is 1.
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Bernoulli Trials Example

• Probability that Mary hits target p1/4,
• Mary fires 6 times, n6,
• What is the probability Mary hits the target more
  than 4 times?

In EXCEL (6)((1/4)5)((3/4)1)(1/4)6
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Random variables and probability distributions.

• Suppose you toss a coin two times. There are four
  possible outcomes HH, HT, TH, and TT. Let the
  variable X represents the number of heads that
  result from this experiment. The variable X can
  take on the values 0, 1, or 2. In this example, X
  is a random variable because its value is
  determined by the outcome of a statistical
  experiment.

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Random variables and probability distributions.

• A probability distribution is a table or an
  equation that links each outcome of a statistical
  experiment with its probability of occurrence.
  The table below, which associates each outcome
  (the number of heads) with its probability. This
  is an example of a probability distribution.

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Random Variable

• A random variable X on a finite sample space S is
  a function (or mapping) from S to a number R in
  S.
• Let S be sample space of outcomes from tossing
  two coins. Then mapping a is
• SHH,HT,TH,TT (assume HT?TH)
• Xa(HH)1, Xa(HT)2, Xa(TH)3, Xa(TT)4
• The range (image) of Xa is
• S1,2,3,4

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Random Variable

• Let S be sample space of outcomes from tossing
  two coins, where we are interested in the number
  of heads. Mapping b is
• SHH,HT,TH,TT
• Xb(HH)2, Xb(HT)1, Xb(TH)1, Xb(TT)0
• The range (image) of Xb is
• S0,1,2

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Random Variable

• A random variable is a function that maps a
  finite sample space into to a numeric value. The
  numeric value has a finite probability space of
  real numbers, where probabilities are assigned to
  the new space according to the following rule
• pointi P(xi) sum of probabilities of points
  in S whose range is xi.
• Recall function F Domain -gt Range (Image)

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Random Variable

• The function assigning pi to xi can be given as a
  table called the distribution of the random
  variable.
• pi P(xi)
• number of points in S whose image is xi
• number of points in S
• (i 1,2,3...n) gives the distribution of X

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Random Variable

• The equiprobable space generated by tossing pair
  of fair dice, consists of 36 ordered pairs(1)
• Slt1,1gt,lt1,2gt,lt1,3gt...lt6,6gt
• Let X be the random variable which assigns to
  each element of S the sum of the two dice
  integers 2,3,4,5,6,7,8, 9,10,11,12

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Random Variable

• Continuing with the sum of the two dice.
• There is only one point whose image is 2, giving
  P(2)1/36.
• There are two points whose image is 3, giving
  P(3)2/36. (lt1,2gt?lt2,1gt, but their sums are )
• Below is the distribution of X.

36/36
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Example Random Variable

• A box contains 9 good items and 3 defective items
  (total 12 items). Three items are selected at
  random from the box. Let X be the random variable
  that counts the number of defective items in a
  sample. X has a range space Rx
  0,1,2.3.
• The sample space 12-choose-3 220 different
  samples of size 3.
• There are 9-choose-3 84 samples of size 3 with
  0 defective items.
• There are 3 9-choose-2 108 samples of size
  3 with 1 defective.
• There are 3-choose-2 9 27 samples of size 3
  with 2 defective.
• There 3-choose-3 1 samples of size 3 with 3
  defective items.
• Where n-choose-r means the number of
  combinations

COMBIN(12,3))
84 108 27 1 ----- 220
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Example Random Variable

• A box contains 9 good items and 3 defective items
  (total 12 items). Three items are selected at
  random from the box. Let X be the random variable
  that counts the number of defective items in a
  sample. X can have values 0-3.
• Below is the distribution of X.

84 108 27 1 ----- 220
220/220
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Functions of a Random Variable

• If X is a random variable then so is Yf(X).
• P(yk) sum of probabilities xi, such that
  ykf(xi)

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Expectation and variance of a random variable

• Let X be a discrete random variable over sample
  space S.
• X takes values x1,x2,x3,... xt with respective
  probabilities p1,p2,p3,... pt
• An experiment which generates S is repeated n
  times and the numbers x1,x2,x3,... xt occur with
  frequency f1,f2,f3,... ft (?fin)
• If n is large then
• one expects

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Expectation of a random variable

• So becomes
• The final formula is the population mean,
  expectation, or expected value of X is denoted as
  ? or E(X).

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Variance of a random variable

• The variance of X is denoted as ?2 or Var(X).
• 2
  2
• The standard deviation is

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Expected value, Variance, Standard Deviation

• E(X) µ µx ??xipi
• Var(X) ?2 ?2x ?(xi - µ)2pi
• SD(X) ?x

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Relation between population and sample mean.

• If we select a sample size N at random from a
  population, then it is possible to show that the
  expected value of the sample mean m is the
  population mean µ.
• This rule differs slightly for variance. The
  sample variance is (N-1)/N times the population
  variance.

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Example Random Variable Expected Value

• A box contains 9 good items and 3 defective
  items. Three items are selected at random from
  the box. Let X be the random variable that counts
  the number of defective items in a sample. X can
  have values 0-3.
• Below is the distribution of X.

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Example Random Variable Expected Value

• µ is the expected value of defective items in
  in a sample size of 3.
• µE(X)
• 0(84/220)1(108/220)2(27/220)3(1/220)132/220
  ?
• Var(X)
• 02(84/220)12 (108/220)22 (27/220)32 (1/220)
  - µ 2 ?
• SD(X) sqrt(µ2)?

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Fair Game1?

• If a prime number appears on a fair die the
  player wins that value. If an non-prime appears
  the player looses that value. Is the game
  fair?(E(X)0)
• S1,2,3,4,5,6
• E(X) 2(1/6)3(1/6)5(1/6)(-1)(1/6)(-4)(1/6)(-
  6)(1/6) -1/6
• Note 1 is not prime

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Fair Game2?

• A player gambles on the toss of two fair coins.
  If 2 heads occur the player wins 2 Euro. If 1
  head occurs he wins 1 Euro. If no heads occur he
  looses 3 Euro. Is the game fair?(E(X)0)
• SHH,HT,TH,TT,
• X(HH) 2, X(HT)X(TH)1, X(TT)-3
• E(X) 2(1/4)1(2/4)-3(1/4) 0.25

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Mean(µ), Variance(?2), Standard Deviation(?)
xi 2 3 11
pi 1/3 1/2 1/6
µExipi 2(1/3) 3(1/2) 11(1/6) 4 E(X2)
Exipi 2(1/3) 3(1/2) 11(1/6) 26 ?2 Var(X)
E(X2) µ2 26 42 10 ? sqrt(Var(X))
sqrt(10) 3.2
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Mean(µ), Variance(?2), Standard Deviation(?)
xi 2 3 11
pi 1/3 1/2 1/6
µExipi 2(1/3) 3(1/2) 11(1/6) 4 E(X2)
Exipi 2(1/3) 3(1/2) 11(1/6) 26 ?2 Var(X)
E(X2) µ2 26 42 10 ? sqrt(Var(X))
sqrt(10) 3.2
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Distribution Example(1)

• Five cards are numbered 1 to 5. Two cards are
  drawn at random .Let X denote the sum of the
  numbers drawn. Find (a) the distribution of X and
  (b) the mean, variance, and standard deviation.
• There are C(5,2) 10 ways of drawing two cards
  at random.

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Distribution Example(2)

• Ten equiprobable sample points with their
  corresponding X-values are

points 1,2 1,3 1,4 1,5 2,3 2,4 2,5 3,4 3,5 4,5
xi 3 4 5 6 5 6 7 7 8 9
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Distribution Example(3)

• The distribution is

xi 3 4 5 6 5 6 7 7 8 9
pi 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1
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Distribution Example(4)

• The distribution is

xi 3 4 5 6 5 6 7 7 8 9
pi 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1

• The mean is 3(0.1)....9(0.1)6
• The E(X2) is 32(0.1)....92(0.1) 39
• The variance is 39 62 3
• The SD is sqrt(3) 1.7