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Introduction |
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Suppose the two events are not independent, that is the occurrence of one depends on the occurrence of other, then how do we compute  This can be explained by conditional probability. |
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Baye's theorem is named after the British mathematician Thomas Bayes who published it in a research paper in 1763. It gives one of the important applications of the conditional probabilities by using the additional information supplied by the experiment or the past records. |
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Conditional Probability |
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Let A and B be any two events associated with a random experiment. The probability of occurrence of event A when the event B has already occurred is called the conditional probability of A when B is given and is denoted as P(A/B). |
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Baye's Theorem |
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Law of Total Probability: If B1, B2, B3, ….., Bn are mutually exclusive and exhaustive events of the sample space S, then for any event A of S. |
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Baye's Theorem: Let S be a sample space. If A1, A, A3 ... An are mutually exclusive and
exhaustive events such that P(Ai) ?
0 for all i. Then for any event A which is a subset of
We have,
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Random Variable and Probability Distribution |
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Let S be a sample space associated with a given random experiment. |
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A real valued function X which assigns to each wi
Î S, a unique real number, X(wi)
= xi is called a random variable. Two types of random variables are 1. Continuous random variable, 2. discrete random variable. |
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Let X be a discrete random variable which takes values x1, x2, x3,…xn where pi = P{X = xi} |
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Then X : x1 x2 x3 …xn |
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P(X) : p1 p2 p3 … pn is called the probability distribution of x. |
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Binomial Distribution |
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A trial, which has only two outcomes i.e., "a success" or "a failure", is called a Bernoulli trial. |
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The probability distribution of the number of successes, so obtained is called the binomial distribution. |
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Poisson Distribution |
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Poisson distribution is a limiting process of binomial distribution. Poisson distribution occurs when there are events which do not occur as outcomes of a definite number of outcomes. |
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Poisson distribution is used under the following conditions: |
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1. Number of trials n tends to infinity |
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2. Probability of success p tends to zero and |
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3. np = l is finite. |
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Applications |
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Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation where it is called "pathway analysis". |
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Example: In a housing colony 70% of the houses are well planned and 60% of the houses are well planned and well built. Find the probability that an arbitrarily chosen house in this colony is well built given that it is well planned. |
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Summary |
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1. If x is a discrete random variable assuming the values x1, x2, x3,….,xn with probabilities p1, p2, p3,…., pn respectively then (x1,p1), (x2, p2),…(xn, pn) defines a probability distribution of X. |
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2. Let n independent bernoulli trials be performed and X denote the number of successes in n trials then X follows a binomial distribution. |
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3. The probability of success be p,
The probability of failure = 1- p = q
The probability of r success in n trials
= P {X = r} = nCr pr qn-r is called the probability mass function of the binomial distribution. |
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4. Mean of binomial distribution = E(X) = np. |
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5. Variance of binomial distribution = V(X) = npq. |
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Conclusion |
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In this chapter we have studied the method of evaluating probabilities of events relating to independent events and conditional events. |
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We have also studied about random variables and their probability distributions, namely binomial distribution and Poisson distribution.
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