Continuous probability distributions

1
Continuous probability distributions

• Uniform probability distribution (ASW, section
  6.1)
• Normal probability distribution (ASW, section
  6.2)
• Table Appendix B and inside front cover
• Excel Appendix 6.2
• Bring the text to class on Wednesday, October 1.
  We will be using Table 1 of Appendix B of ASW.

Notes for October 1, 2008
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Probability density function

• For a continuous random variable x, the
  probability that x takes on a specific value is
  zero. As a result, probabilities for values of
  x are assigned across an interval of values of x.
  The probability function f(x) that is used to
  assign these probabilities is termed the
  probability density function.
• If a variable x has probability density function
  f(x), the probability that x takes on values
  between a and b is the area under the graph of
  f(x) that lies between a and b (or the integral
  of f(x) over the range a to b).

3
Uniform probability distribution (ASW, 226)

• This is a distribution where the probability
  density function f(x) has the same value across
  all the values of x for which it is non-zero.

4
Diagram of a uniform probability distribution

• If x represents the variable being considered,
  the distribution has density f(x) 1/(b a)
  over the range from a to b and density of 0
  elsewhere.

f(x)
1/(b-a)
x
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Example cost of travel

• Suppose a firm reimburses employees at the rate
  of 40 cents per kilometre when an employee uses
  his or her own automobile for company travel.
  Over the past years, the number of kilometres
  reimbursed in this manner has been between
  100,000 and 150,000. The probability
  distribution of anticipated annual travel costs
  for the firm is considered to be a uniform
  distribution over this range.
• At the lower bound of 100,000 km., the total cost
  would be 40,000 and at the upper bound of
  150,000 km., the total cost would be 60,000.
  Since travel is not anticipated to be less than
  100,000 km. nor greater than 150,000 km., cost is
  zero outside the lower and upper bound.

6
Uniform probability distribution for travel
example
Let x be the anticipated total cost of annual
travel for the firm in thousands of dollars.
Then the uniform probability distribution is
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Uniform probability distribution of total
expected travel cost for the firm

• Let x represents the total anticipated annual
  travel costs in thousands of dollars, the
  distribution has density f(x) 1/(60 - 40)
  1/20 over the range from 40 to 60 and density of
  0 elsewhere.

f(x)
1/20
x
Travel cost in thousands of dollars
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Area of the distribution

• Note that the total area under the density
  function f(x) between 40 and 60 equals 1.
• Area height x length (1/20) x (6040)
  (1/20) x 20 1 and this equals the probability
  that some value within the range from 40 to 60
  occurs.

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Area and probability over an interval

• What is the probability that travel costs are
  between 40 and 50?
• Area under the curve between 40 and 50
• height x length (1/20) x (5040) (1/20) x
  10 0.5
• and this is the required probability.

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Expected value and variance for a uniform
probability distribution
For the travel cost example E(x) (40 60)/2
50 Var(x) (6040)2/12 33.333 Standard
deviation of x is the square root of 33.333 or
5.774
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Firms anticipated travel costs

• If the uniform distribution applies, then the
  firm might predict that travel costs will be
  50,000 annually (the mean or expected value).
  However, there is variability in anticipated
  costs, with the standard deviation being 5,774.
  
• The probability that travel costs are within one
  standard deviation of the mean of 50,000 turns
  out to be 0.5774. This is the area under the
  line between 50,000 - 5,774 44,226 and 50,000
  5,774 55,774. This is a distance of 11,548.
  The area under the line is (11,548/20,000) 0.
  5774.
• Note that the area under the curve within two
  standard deviations of the mean is the whole
  distribution.

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Features of a continuous probability distribution
ASW (228)

• For a continuous probability distribution, the
  probability for the random variable must be
  defined over an interval, not at a single value
  of the variable.
• The probability that the random variable takes on
  values within an interval is the area under the
  curve of the density function f(x) across that
  interval. (Or it equals the integral of f(x)
  from the lower to upper bounds of the interval).
• Also note that the total area under the curve of
  the density function equals one.
• Most continuous distributions do not have the
  linear or straight line characteristic of the
  uniform distribution, but will be nonlinear or
  curved. Tables of these distributions are often
  available. These tables give the required areas
  or probabilities.

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Normal probability distribution

• The normal probability distribution is the most
  common and important of the continuous
  probability distributions used in statistical and
  econometric work.
• Other names for the normal distribution are the
  bell curve, since it has a sort of bell shape,
  and the Gaussian distribution, after Gauss, who
  is considered to be the first to have described
  and used the distribution.

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Formula and parameters for the normal distribution

• There are many normal distributions, but any
  normal distribution can be described and graphed
  with two parameters (µ and s) and the following
  formula.

where µ is the mean of the normal
distribution s is the standard deviation of the
normal distribution p is 3.14159 e is 2.71828,
the base of the natural logarithms
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Some characteristics of the normal distribution

• The curve is entirely described by µ, the mean,
  and s, the standard deviation, using the formula
  above.
• The curve peaks at the mean, µ, so the mode also
  equals µ.
• The distribution is symmetric about the centre,
  µ, so the median is also µ. The distribution is
  not skewed.
• The tails of the distribution never quite reach
  the horizontal axis, but get closer and closer to
  this axis the further away from the centre x is.
  This characteristic means that the distribution
  is said to be asymptotic to the horizontal axis.
• The probability that a normally distributed
  variable x takes on values in the range from a to
  b is the area under f(x) between a and b.
• The total area under the curve is 1 the area
  under the curve to the left of centre is 0.5 and
  the area right of centre is 0.5.

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Reasons for using the normal distribution

• Describes some characteristics of populations.
  Eg. Height, weight, and perhaps weight of
  packaged foods and travel time to work. Some
  consider intelligence and ability to be normally
  distributed. Grades for a large number of
  students across classes are often normally
  distributed.
• Characteristics such as incomes, wealth, assets
  and debts, farm size, and stock prices are
  usually not normal. But it is sometimes possible
  to transform these to the normal.
• The normal provides an approximation to
  probabilities such as the binomial when n is
  large, is the limiting distribution of the t
  distribution, and forms the basis for other
  distributions.
• Many statistics obtained from random samples have
  a normal distribution. In particular, when n is
  large, the sample means from randomly selected
  samples haves a normal distribution (ASW, 271).

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Standard normal distribution (z)

• Each µ and s define a different normal
  distribution for a variable x.
• But any normally distributed variable can be
  transformed into the standard normal variable
  (and vice-versa).
• The standard normal variable has a mean of zero
  and a standard deviation of 1 and is usually
  referred to as z.
• Any normally distributed variable x can be
  transformed into the standard normal variable z
  by using the transformation
• The inverse transformation is

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Some probabilities for z

• P(z lt -1) 0.1587
• P(z gt 1) 1 0.8413 0.1587
• P(z lt -1.57) 0.0582
• P(z gt 0.43) 1 0.6664 0.3336
• P (-1.37 lt z lt 1.75) 0.9599 0.0853 0.8746
• P (1.32 lt z lt 2.36) 0.9909 0.9066 0.0843
• P (-1 lt z lt1) 0.8413 0.1587 0.6826
• P (-2 lt z lt 2) 0.9772 0.0228 0.9544

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z values for areas

• Area of 0.05 in the right tail of the
  distribution is obtained by finding the z where
  the cumulative probability reaches 1 - 0.05
  0.95, that is, at z 1.64 or z 1.65. For this
  area, z 1.645 is often used.
• Area of 0.025 in each tail of the distribution,
  or a total of 0.05 in the two tails. The
  cumulative probability first reaches 0.025 at z
  -1.96. By symmetry, the z value in the right
  tail is a 1.96. The interval (-1.96, 1.96)
  contains 95 of the distribution leaving a total
  of 5 in the two tails of the distribution.
• Total area of 0.01 in the two tails is given by
  the area to the left of z -2.575 and to the
  right of z 2.575.
• The above z values will be used extensively later
  in the semester.

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Normal distribution of grades?

Grade (x) Per cent of grades
lt50 7.5
50-60 16.3
60-70 26.6
70-80 30.0
80-90 16.6
90 3.0
Total 100.0
For this distribution, µ 69 and s 14
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Calculations for two intervals of grade
distribution

• 1. Grade less than x 50?
• z (x-µ)/s (50 69)/14 -1.36 and the
  cumulative probability is 0.0869. If exactly
  normal, 8.7 of grades would be less than 50,
  whereas 7.5 actually were less than 50.
• 2. Grade of 80 to 90?
• For x 90, z (x-µ)/s (90 69)/14 1.50.
  Cum P 0.9332
• For x 80, z (x-µ)/s (80 69)/14 0.79.
  Cum P 0.7852
• Area between these values is 0.9332 0.7852
  0.1480 or 14.8, which is a little less than the
  16.6 who received grades between 80 and 90.

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Comparing actual and normal distributions

Grade (x) Actual per cent of grades Per cent if normally distributed
lt50 7.5 8.7
50-60 16.3 17.4
60-70 26.6 26.7
70-80 30.0 25.7
80-90 16.6 14.8
90 3.0 6.7
Total 100.0 100.0
And the actual distribution is close to the
normal distribution, especially for grades up to
70. Note that fewer grades of 90 or more were
awarded than if the distribution was exactly
normal.
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If grades are normally distributed with µ 69
and s 14, what grade is required to

• 1. Be in the upper 5 of all grades?
• Upper 5 or 0.05 begins where the cumulative
  probability reaches 1 - 0.05 0.95 and this is
  at z 1.645. Rearranging the formula z
  (x-µ)/s to solve for x gives
• x µ (zs) 69 (1.645 x 14) 69 23.03
  or x 92.
• 2. Not be in the lower 20 for all grades?
• The cumulative probabilities first reach 0.20 at
  z -0.84. Using the same formula as above to
  transform this z into an x gives
• x µ (zs) 69 (-0.84 x 14) 69 11.76
  or x 57.24 and a grade of 58 would ensure that
  one is not in the lower 20 or one-fifth of the
  distribution.

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Additional notes

• Note that the z value is equivalent to the number
  of standard deviations the value of the normal
  variable is from the mean (ASW, 238).
• Most of the distribution is within 3 standard
  deviations or 3 z values of the mean. That is,
  the probability of any normal variable being more
  than 3 z values from the mean is 0.003.
• Excel can be used to obtain normal probabilities.
  See ASW, 255.
• We will study section 6.3 of the text, normal
  approximation of binomial probabilities, when we
  study sections 7.6 and 8.4 of the text.
• Skip section 6.4.

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Next day

• Sampling and sampling distributions ASW,
  chapter 7.
• Begin interval estimation ASW, chapter 8.