# Lesson4 - Continuous Probability Distributions Author Atul...

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Continuous Probability Distributions: Author: Atul Roy Example 1: There are 5 employees in a unit. Random Variable: # of employees present in a unit on a given day 0 1 2 3 4 5 such a random variable is called discrete random variable Example 2: The temperature outside a certain building is somewhere between 0 and 5 degrees C. 0 1 2 3 4 5 The variable can take on any value on the above number line, including the whole number values. such a random variable is called discrete random variable continuous random variable ................................................. 1

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The probabilities for a continuous random variable are given by the areas under a curve given by a continuous density function y f x . such that a) f x 0 for all x. b) The total area under the curve is 1. Example 2: A uniform distribution (anywhere between the two values uniformly) Example 2
y y=f(x) 0 1 2 3 4 5 The above shows a uniform distribution on the interval 0,5 What is the equation y f(x) that is equation of the line shown above? One suggestion is y 1, Total area under the curve, 5 1 1 another is y 5, Total area under the curve, 5 5 1 Bhavin: y 1 5 , that is the density curve is given by f x 1 5 P 2 X 4 2 1 5 0.4 3

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-1 1 2 3 4 5 x y y=f(x) 0 1 2 4 5 1/5 Normal Distribution The density curve of a normal distribution with mean and standard deviation is given by f x 1 2 e x 2 / 2 2 (You do not have to work with these equations, I need these to sketch graphs in these documents) Example 3: 4
Gven that the time that the students take to finish a certain standardized test shows a normal distribution with mean 90 minutes and standard deviation 10 minutes. f x 1 10 2 e x 90 2 / 2 10 2 f x 0 2e-06 4e-06 6e-06 8e-06 1e-05 1.2e-05 y 40 42 44 46 48 50 x Can not see much But recall that the empirical rule suggested that almost the entire data for a normal distribution is within THREE standard deviations of the mean. That is between 90 3 10 60 and 90 3 10 120 minutes 5

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0 0.01 0.02 0.03 0.04 y 50 60 70 80 90 100 110 120 130 x The proportion of students who will finish the exam within 100 minutes is given by the following area 0 0.01 0.02 0.03 0.04 50 60 70 80 90 100 110 120 130 x 6
Ideally it is calculated as (just a commercial for calculus) but we shall work on simpler ways of aclculating this. − 100 f x dx 0.84134 Excel: 7

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Terminology: A normal distribution with the mean 0 and standard deviation 1 is called the standard normal distribution. Standard normal variable is denoted by z. 8
s z 1 2 e z 2 /2 s z 0 0.1 0.2 0.3 0.4 y -4 -2 2 4 z Total area under this curve is − s z dz 1 The area to the right of z 0is0 .5 , The area between z 0 and z 1.76 0 1.76 s z dz 0.46080 In this course, you may use the a standard normal curve areas table in the back of the book 9

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Normal Curve Areas 0 0 . 0
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Lesson4 - Continuous Probability Distributions Author Atul...

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