Basics of Probability

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PROBABILITY

• B.TECH YEAR II
• SEMESTER III
• 2014-2018

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Axioms of Probability

• There are three axioms of probability
• P(A) 0
• P(S) 1
• If (AnB) ?, for this case
• P(AUB) P(A) P(B)

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• Formula
• P(AUB) P(A) P(B) P(AnB)
• P(A/B) P(AnB)/P(B)
• P(B/A) P(AnB)/P(A)
• If A and B are independent
• P(AnB) P(A).P(B)
• P(B/A) P(B)
• P(A/B) P(A)

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• Contd.
• P(A) 1-P(A)
• When ? is a null set , for this case P(?) 0
• If A is the subset of B, for this case P(A)P(B)
• Proof those......

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RANDOM EVENTS

• When we conduct a random experience, we can use
  set notations to describe the possible outcomes
• Let a fair die is rolled, the possible outcome
• S 1,2,3,4,5,6

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RANDOM VARIABLES

• Random variable X(S) is a real valued function of
  the underlying even space s ? S
• A random variable be of two types
• Discrete variable Range finite eg. 0,1,2 or
  infiniteeg. 0,1,2,3.....,n
• Continues variable range is uncountable
  eg.0,1,2,..........n

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Cumulative Distributive Function

• Definition F(x) P(Xx)
• The function is monotonically increasing
• F(-8) 0
• F(8) 1
• P(a lt x b) F(b) F(a)

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Probability Density Function

• Definition p(x) DF(x) here D d/dx
• Properties
• a. p(x) 0
• b. ? p(x)dx 1 limit -8 to 8
• c. P(a lt x b) ? p(x)dx Limit a to b

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Expected Values

• We denote expected values as E(X) and we can
  represent it in two types
• Continues
• E(X) ? x.p(x)dx limit -8 to
  8
• Discrete
• E(X) ? x.p(x) limit -8 to 8

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Variance

• Formula for variance is same for both continues
  and discrete

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Probability Mass Function

• Definition p(x) P(X x)
• Properties
• p(x) 0
• ?p(x) 1 for all values of x
• P(a X b) ?p(x) for all values
  of xa to xb

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Bernoullis Distribution

• Let p(x) 1-p x0
• p x1
• For this case, we can say that
• Mean p
• Variance p(1-p)

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Binomial Distribution

• n represents identical independent trials
• Two possible outcomes, success and failure
• P(success) p, and P(failure) q and pq 1
• Mean np
• Variance npq

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Poisson Distribution

• Suppose that we can expect some independent event
  to occur ? times over a specified time
  interval. The probability of exactly x
  occurrences will be

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Some useful distribution

• Let p(x) when axb
• 0 otherwise
• It is a continues random variable
• So E(X) or mean value
• And variance will be

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Gaussian PDF

• p(x)
• A Gaussian random variable is completely
  determined by its mean and variance

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• Thank You.....