Introduction to statistical concepts (Part II)

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Introduction to statistical concepts(Part II)

• Dr. Yan Liu
• Department of Biomedical, Industrial and Human
  Factors Engineering
• Wright State University

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Probability Concepts and Terms

• Statistical Experiment
• A process that generates a set of results or
  outcomes, each with some possibility of occurring
• Sample Point
• Each of the possible outcomes of the statistical
  experiment
• Sample Space, S
• The set of all possible outcomes of the
  statistical experiment
• Discrete sample space
• A sample space that contains a finite number or a
  countable set
• e.g. the outcomes of dice rolls, drawings from a
  bag of balls with different colors
• Continuous sample space
• A sample space that contains an infinite and
  uncountable set of sample points
• e.g. the outcomes of temperature readings,
  height measurements, and salaries

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Probability Concepts and Terms

• Event
• A subset of a sample space
• May contain some, all, or none of the outcomes
  (sample points) comprising the sample space
• Simple event
• Event that contains only one sample point
• Compound event
• Event that contains two or more sample points
• Null space
• Event that contains no sample points, denoted as Ø

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X the roll of a dice
S ?
S 1, 2, 3, 4, 5, 6
A is a simple event
B is a compound event
C Ø
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Probability Concepts and Terms

• Probability
• The probability of an event A, P(A), is a measure
  of likelihood that the event will actually occur
  (real number between 0 and 1), calculated as the
  fraction of all outcomes in S that belongs to A.
  0 P(A) 1
• Mathematically, P(A) ( of outcomes in A) / (
  of total outcomes in S)

(Eq. 1)
P(S) ?
A spinner has 4 equal sectors colored yellow,
blue, green and red. After spinning the spinner,
what is the probability of landing on each color?
X the color of the spinner after spinning
1. Define X
S yellow, blue, green, red
2. Define S
3. Calculate P(Xyellow), P(Xblue), P(Xgreen),
and P(Xred)
P(Xyellow) ¼
P(Xblue) ¼
P(Xgreen) ¼
P(Xred) ¼
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Probability Concepts and Terms

• Probability (Cont.)
• If an event A has n possible mutually exclusive
  outcomes, A1, A2, An, then P(A) is the
  summation of the probabilities of all the
  outcomes,
• i.e. P(A) P(A1) P(A2) P(An)
• If A1, A2, An are all the possible outcomes in
  S and not two of these can occur at the same
  time, then their probabilities must sum up to 1
  A1, A2, An are said to be collectively
  exhaustive and mutually exclusive

(Eq. 2)
A spinner has 4 equal sectors colored yellow,
blue, green and red. After spinning the spinner,
what is the probability of landing on yellow or
blue?
X the color of the spinner after spinning
P(Xyellow or Xblue) P(Xyellow) P(Xblue)
¼ ¼ ½
yellow, blue, green, red are collectively
exhaustive and mutually exclusive outcomes of X
P(Xyellow) P(Xblue) P(Xgreen) P(Xred)
¼ ¼ ¼ ¼ 1
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Probability Distribution

• The probability distribution of a random variable
  is a function that links each of the values that
  the random variable can assume and its
  probability of occurrence
• Discrete Probability Function (Probability Mass
  Function)
• Probability distribution of a discrete random
  variable
• A function that gives probability p(xi) when the
  random variable X equals xi,
• i.e. For each value xi, P(Xxi) p(xi)
• It must satisfy the following conditions
• a. 0 p(xi) 1
• b. Si p(xi) 1

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A spinner has 4 equal sectors colored yellow,
blue, green and red. X the color of the spinner
after a spinning, what is the probability mass
function of X?
S1
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Roll two dice. X the sum of the rolls of two
dice, what is the probability mass function of X?
X1roll of the 1st die, X2 roll of the 2nd die,
XX1X2
X1 1, 2, 3, 4, 5, 6, X2 1, 2, 3, 4, 5, 6
?
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P(Xxi) ( of outcomes in xi) / ( of total
outcomes in SX)
S36
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Probability mass function of X is as follows
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P(Xx)
x
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Probability Distribution

• Continuous Probability Function (Probability
  Density Function)
• Probability distribution of a continuous random
  variable
• A function that can be integrated to obtain the
  probability that the random variable takes a
  value in a given interval
• For an interval (a, b),
• It must satisfy the following conditions

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Cumulative Probability Distribution

• The cumulative probability distribution of a
  random variable X is a function which gives the
  probability that X is less than or equal to x,
  for every value X x
• FX( X x) P(X x)
• For a discrete variable X with probability mass
  function pX(x),
• its FX(X x) is given by
• For a continuous variable X with probability
  density function f(x), its FX(x) is given by

(Eq. 3)
(Eq. 4)
(Eq. 5)
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Probability mass function of X is as follows
Cumulative probability distribution of X is as
follows
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Expected Value

• The expected value (or population mean) of a
  random variable X, E(X) or µX , is the weighted
  average of all the values that a random variable
  would assume in the long run
• For a discrete variable X, E(X) is calculated as
• For a continuous variable X, E(X) is calculated as

(Eq. 6)
(Eq. 7)
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Probability mass function of X is as follows
E(X) 2(1/36) 3(2/36) 4(3/36) 5(4/36)
6(5/36) 7(6/36) 8(5/36)
9(4/36) 10(3/36) 11(2/36)
12(1/36) 252/36 7
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Variance and Standard Deviation of Random Variable

• The variance of a random variable X, var(X) or
  sX2, is the variance of all the values that the
  random variable would assume in the long run
• Standard deviation of X, sX, is the square root
  of its variance

(Eq. 8)
(Eq. 9)
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Probability mass function of X is as follows
E(X2) 4(1/36) 9(2/36) 144(1/36)
252/36 54.85
var(X) 54.85 72 5.85