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Continuous distributiion-111

Continuous distributiion-111 - Chapter 4 Continuous...

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Chapter 4: Continuous distributions Continuous Random Variables The Uniform Distribution The Gamma Distribution The Normal Distribution The Normal Approximation to Binomial Distribution Exponential Distribution Chi square Distribution
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Rectangular or Uniform distribution A random variable X is said to have a continuous uniform distribution over an interval ( α , β ) if its probability density function is constant k over entire range of x. PROBABILITY DENSITY FUNCTION f (x) = k, α < X < β = 0 otherwise
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Rectangular or Uniform distribution The uniform distribution, with parameters α and β , has probability density function < < - = elsewhere 0 for 1 ) ( β α α β x x f
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1 β α - Figure : Graph of uniform probability density x ( ) f x α β 0 Note : All values of x from α to β are equally likely in the sense that the probability that x lies in an interval of width x entirely contained in the interval from α to β is equal to x /( β - α ), regardless of the exact location of the interval.
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Distribution function for uniform density function < < - - = β β α α β α α x x x x x F for 1 for for 0 ) ( Rectangular or Uniform distribution
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PROBLEM 1 An unprincipled used car dealer sells a car to a buyer, even though the dealer knows that car will have a major breakdown with in the next 6 months.The dealer provides warranty of 45 days on all cars sold.Let X represents the length of time until the breakdown occurs.Assume X is a uniform random variable with values between 0 & 6 months
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PROBLEM 1 Contd a) Graph probability density curve of X b) Calculate the probability that breakdown occurs while the car is still under warranty. c) Plot the graph of cumulative distribution function of X
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The Uniform Distribution (cont’d) Mean of uniform distribution 2 β α μ + = Proof: 2 2 1 1 2 β α α β α β μ β α β α + = - = - = x dx x
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The Uniform Distribution (cont’d) Variance of uniform distribution 2 2 ) ( 12 1 α β σ - = Proof: . 12 ) ( 4 ) ( 3 Hence 3 3 1 1 2 2 2 2 2 2 2 2 2 3 2 2 α β β α α αβ β μ μ σ α αβ β α β α β μ β α β α - = + - + + = - = + + = - = - = x dx x
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The Uniform Distribution (cont’d) Moment generating function dx e dx x f e t M tx tx X - = = β α β α α β / 1 ) ( ) ( ) /( ) ( ) ( α β α β - - = t t X e e t M
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PROBLEM 2 Subway train on a certain line run every half hour between midnight & six in the morning. What is the probability that a man entering the station at a random time during this period will have to wait at least 20 minutes.
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Discrete Uniform distribution If random variable assume finite no. of values with each value occuring with same probability Probability density function is f(x) = 1/n, X=x 1 ,x 2 ,…… x n
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PROBLEM 3 The manager of a local soft drink bottling company believes when a new beaverage dispensing machine is set to dispense 7 ounces, it infact dispense an amount of X at random anywhere between 6.5 & 7.5 ounces inclusive.
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