The Uniform Distribution

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The Uniform Distribution
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Parameters

• Parameters describe populations and probability
  distributions. The first parameter specified here
  is shape uniform. For a uniform variable the min
  and max are then usually given as parameters.

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Parameters

• Range b a

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Parameters

• a and b are given a is minimum b is maximum
• The mean and median are obvious (a b)/2.

1/(b a) is height
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Parameters

• Range b a

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Finding Probabilities Percentiles

• Probabilities and percentiles are found easily
  all areas under the curve involve rectangles.

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Example

• A company ships packages with weights that vary
  according to a Uniform distribution with minimum
  1 pound and maximum 6 pounds.

Height 1/5 0.20 Range 6 1 5
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Example

• Determine the median, mean and standard deviation.

? Mean (a b) / 2 (1 6) / 2 3.5
lbs Median 3.5
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Example

• Determine the median, mean and standard deviation.

? Mean (a b) / 2 (1 6) / 2 3.5
lbs Median 3.5
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Example

• Determine the median, mean and standard deviation.

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Example

• What is the probability a package weighs more
  than 2.75 pounds?

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Example

• What is the probability a packages weighs more
  than 2.75 pounds?

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Example

• What is the probability a packages weighs more
  than 2.75 pounds?

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Example

• What is the probability a packages weighsmore
  than 2.75 pounds?
• Find the
• shaded area.

Width 6 2.75 3.25 Height 0.20 Area 3.25 ?
0.20 0.65
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Example

• What is the probability a packages weighs more
  than 2.75 pounds? 0.65
• 65.0 of packages weigh more than 2.75 lbs.

2.75 lbs is the 35th percentile of the package
weights.
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Example

• Find the 23rd percentile of the package weights.
• Probability Area.

We know the probability (0.23) we need the
weight.
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Example

• Find the 23rd percentile of the package weights.
• Probability Area.

Width ? Height Area Width ? 0.20 0.23 Width
0.23 / 0.20 1.15 1.15 above 1 1 1.15 2.15
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Example

• Find the 23rd percentile of the package weights.
• 2.15. 23 of packages weigh less than 2.15 lbs.