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Continuous Probability Distribution Functions The probability distribution below shows a binomial distribution with n = 6 and p = 0.4 The width of each bar is one and the height is the probability that each x value is obtained : so the total area =1 0.04665 0.18662 0.31104 0.27648 0.13824 0.036864 0.004096 Binomial Graph

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Continuous Probability Distribution Functions 0.04665 0.18662 0.31104 0.27648 0.13824 0.036864 0.004096 The x values can only take discrete values : 0, 1, 2, 3, 4, 5, 6 For a continuous probability function x can take any value so the bars are infinitely thin

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So as the gap between the x values gets smaller the histogram starts to become a curve So now the area under the curve = 1 To find the probability that x lies between 2 values we need to find the area between the 2 values So P(a < x< b) = ie. area under curve y = f(x) between ‘a‘ and ‘b‘ The function f(x) is called a Probability Density Function – P.D.F

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Ex1 The waiting time for a London tube is a maximum of 10 mins Find the P.D.F and the probability that the waiting time >7mins As the area = 1 the height of the rectangle must equal So P.D.F is f(x) = X = waiting time P(X > 7) = area from x = 7 to 10 = 3 = 0.3 Uniform distribution Probability Density Function – P.D.F

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Note total probabilities add upto 1

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