Download Defining Probabilities: Random Variables

Defining Probabilities: Random Variables
• Examples:
– Out of 100 heart catheterization procedures performed
at a local hospital each year, the probability that more
than five of them will result in complications is
__________
– Drywall anchors are sold in packs of 50 at the local
hardware store. The probability that no more than 3 will
be defective is
__________
– In general,
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___________
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Discrete Random Variables
• Example:
– Look back at problem 2.53, page 55. Assume
someone spends $75 to buy 3 envelopes. The
sample space describing the presence of $10 bills (T)
vs bills that are not $10 (N) is:
_____________________________
– The random variable associated with this situation, X,
reflects the outcome of the choice and can take on
the values:
_____________________________
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Discrete Probability Distributions
• The probability that there are no $10 in the
group is
P(X = 0) = ___________________
The probability distribution associated with the
number of $10 bills is given by:
x
0
1
2
3
P(X = x)
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Another Example
• Example 3.8, pg 80
P(X = 0) =
_____________________
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Discrete Probability Distributions
• The discrete probability distribution function (pdf)
– f(x) = P(X = x) ≥ 0
– Σx f(x) = 1
• The cumulative distribution, F(x)
– F(x) = P(X ≤ x) = Σt ≤ x f(t)
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Probability Distributions
• From our example, the probability that no more
than 2 of the envelopes contain $10 bills is
P(X ≤ 2) = F(2) = _________________
• The probability that no fewer than 2 envelopes
contain $10 bills is
P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - F(1) = ________________
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Another View
• The probability histogram
0.45
0.4
0.35
f(x)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
x
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Your Turn …
• The output from of the same type of circuit board from two
assembly lines is mixed into one storage tray. In a tray of 10 circuit
boards, 6 are from line A and 4 from line B. If the inspector chooses
2 boards from the tray, show the probability distribution function
associated with the selected boards being from line A.
x
P(x)
0
1
2
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Continuous Probability Distributions
• Examples:
– The probability that the average daily temperature in
Georgia during the month of August falls between 90
and 95 degrees is
__________
– The probability that a given part will fail before 1000
hours of use is
__________
– In general,
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__________
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Understanding Continuous
Distributions
• The probability that the
average daily
temperature in Georgia
during the month of
August falls between 90
and 95 degrees is
-5
-3
-1
1
3
5
• The probability that a
given part will fail before
1000 hours of use is
0
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10
15
20
25
30
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Continuous Probability Distributions
• The continuous probability density function (pdf)
f(x) ≥ 0, for all x ∈ R

 f ( x )dx  1

b
P (a  X  b)   f ( x )dx
a
• The cumulative distribution, F(x)
x
F ( x )  P ( X  x )   f (t )dt

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Probability Distributions
• Example: Problem 3.7, pg. 88
f(x) =
{
x,
2-x,
0,
0<x<1
1≤x<2
elsewhere
1st – what does the function look like?
a) P(X < 120) = ___________________
b) P(50 < X < 100) = ___________________
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Your turn
• Problem 3.14, pg. 89
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