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Unformatted text preview: lecture 10, Continuous Random Variables I Continuous Probability Distributions The Uniform Distribution lecture 10, Continuous Random Variables I Outline Continuous Probability Distributions The Uniform Distribution • The Normal Distribution • The Normal Approximation lecture 10, Continuous Random Variables I Continuous Probability Distributions The Uniform Distribution lecture 10, Continuous Random Variables I Continuous Probability Distributions Continuous Probability Distributions A continuous random variable may assume any numerical value in one or more intervals • Eg: portfolio value by the end of the year, finish time of the winning horse in a horse racing, bus waiting time,... • How to describe their probability distributions? For example, what’s the probability that the finish time is exactly 1:47:47? – This probability is 0! • Use a continuous probability distribution to assign probabilities to intervals lecture 10, Continuous Random Variables I Continuous Probability Distributions The Uniform Distribution lecture 10, Continuous Random Variables I Continuous Probability Distributions Continuous Probability Distributions ( ctd) • A curve f ( x ) is the continuous probability distribution of the continuous random variable X if the probability that X will be in a specified interval is equal to the area under the curve f ( x ) corresponding to the interval • Other names for a continuous probability distribution: • Probability curve • Probability density function, or simply density function lecture 10, Continuous Random Variables I Continuous Probability Distributions The Uniform Distribution lecture 10, Continuous Random Variables I Continuous Probability Distributions Area and Probability • For any a < b , the blue area under the curve f ( x ) from x = a to x = b is the probability that X could take any value in the range a to b — denoted by P ( a ≤ X ≤ b ) • P ( a ≤ X ≤ b ) = P ( a < X < b ) = P ( a ≤ X < b ) = P ( a < X ≤ b ) , because each of the interval endpoints has a probability of 0 lecture 10, Continuous Random Variables I Continuous Probability Distributions The Uniform Distribution lecture 10, Continuous Random Variables I Continuous Probability Distributions Properties of Continuous Probability Distributions • Properties of f ( x ) : f ( x ) is a (usually continuous) function such that • f ( x ) ≥ 0 for all x • The total area under the curve of f ( x ) is equal to 1 • Essential point: An area under a continuous probability distribution is a probability • Is f ( x ) equal to P ( X = x ) ? NO! For continuous rv, P ( X = x ) = 0, but f ( x ) can even be bigger than 1! lecture 10, Continuous Random Variables I Continuous Probability Distributions The Uniform Distribution lecture 10, Continuous Random Variables I Continuous Probability Distributions Recall, Histogram (Lect 2) • area of each rectangle in the histogram equals the corresponding relative frequency of that class....
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 Fall '09
 AnthonyChan
 Normal Distribution, Probability distribution, probability density function

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