maths

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maths statistics 3. Distributions Dr
William Megill 2E2.31 enswmm_at_bath.ac.uk
ME10305
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Stochasticity

• Reality not always well behaved
• random effects introduce noise
• All measurements subject to random variation
• No matter how careful you might be
• usually small
• sometimes big enough to mask conclusions
• signal to noise ratio
• Though models need not be perfect
• e.g. Newtons laws are good enough for most
  purposes
• Need a means of testing model applicability

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Random Variables

• Definition
• A numerical variable whose measured value can
  change from one replicate of an experiment to
  another
• Types
• Discrete finite set of real numbers
• e.g. scratches, defective parts, bits in
  transmission
• Continuous interval of real numbers
• e.g. current, length, pressure, temp, time, V,
  mass

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Probability Distributions

• Recall Dr. Reess lectures on probability
• ProbDist definition
• Description of the set of the probabilities
  associated with the possible values for X.
• Probability Density Function
• Not unlike density functions used elsewhere in
  Eng.
• e.g. density of a beam as function of length
• given by
• properties

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Probability Distributions

• Area under portion of the curve is probability of
  that interval
• e.g. histogram
• How to use?
• Given a distribution of current measurements,
  what is the probability that the current
  measured this time is less than 10mA?

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Summary Statistics

• Mean balance point
• Variance

E(x) expected value
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Calculation formulae

• Mean
• Standard Deviation

42.4 65.7 29.8 58.7 52.1 55.8 57.0 68.7 67.3 67.3
54.3 54.0
1797.76 4316.49 888.04 3445.69 2714.41 3113.64 3
249 4719.69 4529.29 4529.29 2948.49 2916
numerator 39168-37755 1412
variance 1412 / 11 128.4
stdev sqrt(128.4) 11.3
Sum 673.1
Sum squares 39167.79
Mean 56.1
over n 37755
Sum squared 453063
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Normal Distribution

• History
• Discovered by DeMoivre (1733)
• Lost until Laplace (1812)
• Independently discovered by Gauss (1830)
• Probability density function
• Std norm variable
• if X normal rand var, with E(X)m, V(X)s2

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Normal Distribution

• Area under whole curve 1
• Probability of ?
• Area under curve between Z0 and Z0.54
• Use formula or lookup table
• p(0ltZlt0.54)0.2054
• linear interpolation
• Careful to read table caption
• Here proportion of curve between zero and Z
• Often given as proportion that lies beyond Z
• Important values
• p(0ltZlt0.955) 0.33
• p(0ltZlt1.96) 0.4750
• p(0ltZlt2.575) 0.495

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Normal Distribution

• Add/subtract areas to calculate other
  probabilities

prob that Zgtz1 and Zltz2
p(0ltZltz2)
p(0ltZltz1)
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Central Limit Theorem

• Definition
• Whenever a random experiment is replicated, the
  random variable that equals the average result
  over the replicates tends to have a normal
  distribution as the number of replicates becomes
  large