CONTINUOUS RANDOM VARIABLES

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• CONTINUOUS RANDOM VARIABLES

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CONTINUOUSRANDOM VARIABLES

• Continuous random variables have values in a
  continuum of real numbers
• Examples --
• X How far you will hit a golf ball
• Y How many hours you will spend studying
• Z The weight of a watermelon
• W Third quarter profits of a company

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POINT PROBABILITY

• What is the probability you hit a golf ball
  exactly 156.73567263541 yards, 156.73 yards?, 156
  yards?
• For continuous random variables
• P(X any specific ) 0

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PROBABILITY DENSITY

• What is the probability you hit a golf ball
  between 100 and 200 yards, between 250 and 350
  yards, between .?
• Probability density f(x) is a measure (it is not
  a probability) of the likelihood you will get a
  value around x

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PROPERTIES OF PROBABILITY DENSITY FUNCTIONS f(x)
f(x) ? 0 for all values of x

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INTERVAL PROBABILITIES

• P(a ? X ? b)
• Area under the curve between a and b

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EXAMPLE

• If f(x) .375x2 for (0ltxlt2), find P(1?X?1.5)

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EXPECTED VALUE

• DISCRETE
• E(X) ? ? ?xp(x) over all values of x
• CONTINUOUS
• Replace

p(x) with f(x)dx
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Mean -- Sample Calculation

• f(x) .375x2 for 0ltXlt2

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VARIANCE
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Example -- Variance

• f(x) .375x2 for 0ltXlt2

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Approaches to CalculatingContinuous Probabilities

• Probabilities are areas under density functions
• These areas are found using integral calculus
• Fortunately, the results for the most common
  continuous distributions can be found using
• Tables
• Excel

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REVIEW

• Continuous random variables are those that can
  assume any value in a continuous interval
• They are described by probability density
  functions
• Probabilities are areas under the density curve
• Means and variances are calculated the same way
  as that for discrete random variables except that
  p(x) is replaced by f(x)dx and the summation sign
  is replaced by the integral sign