Chapter8_STAT1100.ppt", filename="Chapter8_STAT1100.ppt", filename="Chapter8_STA

# Chapter8_STAT1100.ppt", filename="Chapter8_STAT1100.ppt", filename="Chapter8_STA

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Continuous Probability Distributions Statistics for Management and Economics Chapter 8

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Probability Density Functions Unlike a discrete random variable which we studied in Chapter 7, a continuous random variable is one that can assume an uncountable number of values. We cannot list the possible values because there is an infinite number of them. Because there is an infinite number of values, the probability of each individual value is virtually 0.
Point Probabilities are Zero Because there is an infinite number of values, the probability of each individual value is virtually 0. Thus, we can determine the probability of a range of values only. E.g. with a discrete random variable like tossing a die, it is meaningful to talk about P(X=5), say. In a continuous setting (e.g. with time as a random variable), the probability the random variable of interest, say task length, takes exactly 5 minutes is infinitesimally small, hence P(X=5) = 0. It is meaningful to talk about P(X ≤ 5).

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Probability Density Function A function f(x) is called a probability density function (over the range a ≤ x ≤ b if it meets the following requirements: 1) f(x) ≥ 0 for all x between a and b , and 2) The total area under the curve between a and b is 1.0 f(x) x b a area=1
Characteristics of Density Functions All density functions must satisfy the following two requirements: 1. The curve must never fall below the horizontal axis. That is, f (x) ≥ 0 for all x 2. The total area between the curve and the horizontal axis must be 1. In calculus this is expressed as: f(x) dx = 1

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Uniform Distribution Consider the uniform probability distribution (sometimes called the rectangular probability distribution ). It is described by the function: f(x) x b a area = width x height = (b – a) x = 1 *f(x) = 0 for all other values
The Normal Distribution The normal distribution is the most important of all probability distributions. The probability density function of a normal random variable is given by: It looks like this: Bell shaped, Symmetrical around the mean

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The Normal Distribution The normal distribution is fully defined by two parameters: its standard deviation and mean Unlike the range of the uniform distribution (a x b) ≤ ≤ Normal distributions range from minus infinity to plus infinity The normal distribution is bell shaped and symmetrical about the mean
Standard Normal Distribution A normal distribution whose mean is zero

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## This note was uploaded on 06/25/2008 for the course BUSSPP MCE taught by Professor Atkins during the Spring '08 term at Pittsburgh.

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Chapter8_STAT1100.ppt", filename="Chapter8_STAT1100.ppt", filename="Chapter8_STA

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