Download Handout 7 Random Variables

Chapter 8
Random
Variables
Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc.
Random Variables
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An environmental scientist who obtains an air sample from a specified
location might be especially concerned with the concentration of
ozone(a major constituent of smog). Prior to the selection of the air
sample, there is uncertainty as to what value of ozone concentration
will result. Because the value of a variable such as ozone concentration
is subject to uncertainty, it is called a random variable.
A quality control inspector who must decide whether to accept a large
shipment of components may base the decision on the number of
defectives in a group of 20 components randomly selected from the
shipment. The number of defective components among the 20 selected
is subject to uncertainty and thus is called a random variable.
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What is a Random Variable?
A variable whose value depends on the outcome of a
chance experiment is called a random variable.
Two different broad classes of random variables:
1. A continuous random variable can take any value in
an interval or collection of intervals.
2. A discrete random variable can take one of a
countable list of distinct values.
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Example Random Variables at an
Outdoor Graduation or Wedding
Temperature: continuous random variable
Number of airplanes that fly overhead:
discrete random variable
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Continuous Random Variables
Continuous random variable: the outcome can
be any value in an interval or collection of intervals.
Probability density function for a continuous random
variable x is a curve such that the area under the curve over
an interval equals the probability that x is in that interval.
P(a ≤ x ≤ b) = area under density curve over the
interval between the values a and b.
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Example Time Spent Waiting for Bus
Bus arrives at stop every 10 minutes. Person arrives at
stop at a random time, how long will s/he have to wait?
x = waiting time until next bus arrives.
x is a continuous random variable over 0 to 10 minutes.
Note: Height is 0.10
so total area under the
curve is (0.10)(10) = 1
This is an example of a
Uniform random variable
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Example Waiting for Bus (cont)
What is the probability the waiting time x was in the
interval from 5 to 7 minutes?
Probability = area under curve between 5 and 7
= (base)(height) = (2)(.1) = .2
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Example College Women’s Heights
Data suggest the distribution of heights of college
women modeled by a normal curve with mean
μ = 65 inches and standard deviation σ = 2.7 inches.
Note: Tick marks given
at the mean and at 1, 2, 3
standard deviations above
and below the mean.
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Standard Scores
The formula for converting any value x to
a z-score is
Value − Mean
x−μ
z=
=
σ
Standard deviation
A z-score measures the number of standard
deviations that a value falls from the mean.
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Example Height (cont)
For a population of college women, the z-score
corresponding to a height of 62 inches is
z=
x−μ
σ
62 − 65
=
= −1.11
2 .7
This z-score tells us that 62 inches is 1.11 standard
deviations below the mean height for this population.
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Finding Probabilities for z-scores
Table 1 = Standard Normal (z) Probabilities
• Body of table contains P(z ≤ z*).
• Left-most column of table shows algebraic sign,
digit before the decimal place, the first decimal
place for z*.
• Second decimal place of z* is in column heading.
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More Finding Probabilities for z-scores
Table 1 = Standard Normal (z) Probabilities
P(z ≤ -3.00) =.0013 (see in portion above)
P(z ≤ −2.59) = .0048
P(z ≤ 1.31) = .9049
P(z ≤ 2.00) = .9772
P(z ≤ -4.75) = .000001 (from in the extreme)
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Example Probability z > 1.31
P(z > 1.31) = 1 – P(z < 1.31)
= 1 – .9049 = .0951
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Example Probability z is
between –2.59 and 1.31
P(-2.59 ≤ z ≤ 1.31)
= P(z ≤ 1.31) – P(z ≤ -2.59)
= .9049 – .0048 = .9001
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Use z-scores to Solve General Problems
Example Height (cont)
What is the probability that a randomly selected
college woman is 62 inches or shorter?
62 − 65 ⎞
⎛
P ( x ≤ 62 ) = P⎜ z ≤
⎟
2 .7 ⎠
⎝
= P ( z ≤ −1.11) = .1335
About 13% of college women
are 62 inches or shorter.
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Use z-scores to Solve General Problems
Example Height (cont)
What proportion of college woman are taller than 68 inches?
68 − 65 ⎞
⎛
P (x > 68) = P⎜ z >
⎟ = P (z > 1.11) = 1 − P ( z ≤ 1.11)
2 .7 ⎠
⎝
= 1 − .8665 = .1335
About 13% of
college women
are taller than
68 inches.
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Finding Percentiles
If 25th percentile of pulse rates is
64 bpm, then 25% of pulse rates
are below 64 and 75% are above 64.
The percentile is 64 bpm, and the
percentile ranking is 25%.
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Example 75th Percentile of
Systolic Blood Pressure
Blood Pressures are normal with mean 120 and
standard deviation 10. What is the 75th percentile?
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