Uniform Distribution
A package delivery service divides their packages into weight classes. Suppose that
packages in the 14 to 20 pound class are uniformly distributed, meaning that all weights
within that class are equally likely to occur.
X = the weight of a package in the 14 to 20 pound weight class
X~U(14, 20)
Understanding the Picture
The graph of the uniform distribution is a rectangle.
The rectangle starts at 14 and ends at 20.
The height of the rectangle is 1/6 so that the area of the entire rectangle is 1.
Area = 1 = (base)(height)
base = 20−14 = 6
The equation for area becomes 1 = 6h
Solve for height h: h = 1/6
Finding Probabilities
a. Find the probability that a package weighs between 15 and 16.5 pounds.
P(15 ≤ X ≤ 16.5 ) = area = (base)(height)
P(15 ≤ X ≤ 16.5 ) = (16.5-15)(1/6)
P(15 ≤ X ≤ 16.5 )== 1.5 (1/6) = ¼ = 0.25
b. Find the probability that a package weighs less than 15 pounds.
P(X < 15 ) = area = (base)(height)
P(X < 15 ) = (15-14)(1/6)
P(X < 15 )== 1(1/6) = 1/6 = 0.1667
c. Find the probability that a package weighs at least 18 pounds.
P(X ≥ 18 ) = area = (base)(height)
P(X ≥ 18 ) = (20−18)(1/6)
P(X ≥ 18 )== 2(1/6) = 1/3 = 0.3333
Finding the Mean (average) and Standard Deviation
X~U(a, b)
X~U(14, 20)
µ=
a + b 14 + 20
=
= 17
2
2
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σ =
b−a
12
=
20 - 14
12
=
6
12
= 1.732
© R. Bloom 2007
Uniform Distribution
Inverse Problems
A. Finding Percentiles: Percentile are area on the left.
Find the 40th percentile of packages weights in this weight class.
Area = P(X ≤ k) = .40
(base)(height) = .40
(k − 14) (1/6) = .40
(k − 14) = .40(6) = 2.4
k = 14 + 2.4 = 16.4 pounds
P(X ≤ 16.4) = .40
The 40th percentile is 16.4 pounds.
40 % of packages weigh at most 16.4 pounds.
B. Finding X when given area on the right:
20 % of packages weigh at least how much?
Find the minimum weight for the heaviest 20% of packages.
Area = P(X ≥ k) = .20
(base)(height) = .20
(20 − k) (1/6) = .20
(20 − k) = .20(6) = 1.2
−k = −20 + 1.2 = −18.8
k = 18.8 pounds
P(X ≥ 18.8) = .20
20 % of packages weigh at least 18.8 pounds.
What does this mean about the unshaded area on the left?
This means that 80% of packages weigh at most ( ≤ ) 18.8 pounds.
18.8 pounds is the 80th percentile. Percentiles are “area on the left”.
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© R. Bloom 2007