Chapter 5: Continuous Random Variables

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Chapter 5 Continuous Random Variables
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Where Weve Been

• Using probability rules to find the probability
  of discrete events
• Examined probability models for discrete random
  variables

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Where Were Going

• Develop the notion of a probability distribution
  for a continuous random variable
• Examine several important continuous random
  variables and their probability models
• Introduce the normal probability distribution

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5.1 Continuous Probability Distributions

• A continuous random variable can assume any
  numerical value within some interval or
  intervals.
• The graph of the probability distribution is a
  smooth curve called a
• probability density function,
• frequency function or
• probability distribution.

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5.1 Continuous Probability Distributions

• There are an infinite number of possible outcomes
  
• p(x) 0
• Instead, find p(altxltb)
• ? Table
• ? Software
• ? Integral calculus)

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5.2 The Uniform Distribution

• X can take on any value between c and d with
  equal probability
• 1/(d - c)
• For two values a and b

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5.2 The Uniform Distribution

• Mean
• Standard Deviation

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5.2 The Uniform Distribution

• Suppose a random variable x is distributed
  uniformly with
• c 5 and d 25.
• What is P(10 ? x ? 18)?

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5.2 The Uniform Distribution

• Suppose a random variable x is distributed
  uniformly with
• c 5 and d 25.
• What is P(10 ? x ? 18)?

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5.3 The Normal Distribution

• Closely approximates many situations
• Perfectly symmetrical around its mean

• The probability density function f(x)
• µ the mean of x
• ? the standard deviation of x
• ? 3.1416
• e 2.71828

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5.3 The Normal Distribution

• Each combination of µ and ? produces a unique
  normal curve
• The standard normal curve is used in practice,
  based on the standard normal random variable z (µ
  0, ? 1), with the probability distribution

The probabilities for z are given in Table IV
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5.3 The Normal Distribution
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5.3 The Normal Distribution
So any normally distributed variable can be
analyzed with this single distribution

• For a normally distributed random variable x, if
  we know µ and ?,

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5.3 The Normal Distribution

• Say a toy car goes an average of 3,000 yards
  between recharges, with a standard deviation of
  50 yards (i.e., µ 3,000 and ? 50)
• What is the probability that the car will go more
  than 3,100 yards without recharging?

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5.3 The Normal Distribution

• Say a toy car goes an average of 3,000 yards
  between recharges, with a standard deviation of
  50 yards (i.e., µ 3,000 and ? 50)
• What is the probability that the car will go more
  than 3,100 yards without recharging?

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5.3 The Normal Distribution

• To find the probability for a normal random
  variable
• Sketch the normal distribution
• Indicate xs mean
• Convert the x variables into z values
• Put both sets of values on the sketch, z below x
• Use Table IV to find the desired probabilities

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5.4 Descriptive Methods for Assessing Normality

• If the data are normal
• A histogram or stem-and-leaf display will look
  like the normal curve
• The mean s, 2s and 3s will approximate the
  empirical rule percentages
• The ratio of the interquartile range to the
  standard deviation will be about 1.3
• A normal probability plot , a scatterplot with
  the ranked data on one axis and the expected
  z-scores from a standard normal distribution on
  the other axis, will produce close to a straight
  line

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5.4 Descriptive Methods for Assessing Normality
?

• Errors per MLB team in 2003
• Mean 106
• Standard Deviation 17
• IQR 22

?
?
22 out of 30 73
28 out of 30 93
30 out of 30 100
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5.4 Descriptive Methods for Assessing Normality

• A normal probability plot is a scatterplot with
  the ranked data on one axis and the expected
  z-scores from a standard normal distribution on
  the other axis

?
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5.5 Approximating a Binomial Distribution with
the Normal Distribution

• Discrete calculations may become very cumbersome
• The normal distribution may be used to
  approximate discrete distributions
• The larger n is, and the closer p is to .5, the
  better the approximation
• Since we need a range, not a value, the
  correction for continuity must be used
• A number r becomes r.5

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5.5 Approximating a Binomial Distribution with
the Normal Distribution
Calculate the mean plus/minus 3 standard
deviations
If this interval is in the range 0 to n, the
approximation will be reasonably close
Express the binomial probability as a range of
values
Find the z-values for each binomial value
Use the standard normal distribution to find the
probability for the range of values you calculated
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5.5 Approximating a Binomial Distribution with
the Normal Distribution

• Flip a coin 100 times and compare the binomial
  and normal results

BinomialNormal
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5.5 Approximating a Binomial Distribution with
the Normal Distribution

• Flip a weighted coin P(H).4 10 times and
  compare the results

BinomialNormal
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5.5 Approximating a Binomial Distribution with
the Normal Distribution

• Flip a weighted coin P(H).4 10 times and
  compare the results

BinomialNormal
The more p differs from .5, and the smaller n
is, the less precise the approximation will be
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5.6 The Exponential Distribution

• Probability Distribution for an Exponential
  Random Variable x
• Probability Density Function
• Mean µ ?
• Standard Deviation ? ?

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5.6 The Exponential Distribution

• Suppose the waiting time to see the nurse at the
  student health center is distributed
  exponentially with a mean of 45 minutes. What is
  the probability that a student will wait more
  than an hour to get his or her generic pill?

60
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5.6 The Exponential Distribution

• Suppose the waiting time to see the nurse at the
  student health center is distributed
  exponentially with a mean of 45 minutes. What is
  the probability that a student will wait more
  than an hour to get his or her generic pill?

60