Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative...
Transcript of Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative...
![Page 1: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/1.jpg)
Probability and CumulativeDistribution Functions
Lesson 20
![Page 2: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/2.jpg)
RecallIf p(x) is a density function for some characteristic of apopulation, then
![Page 3: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/3.jpg)
RecallIf p(x) is a density function for some characteristic of apopulation, then
We also know that for any density function,
![Page 4: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/4.jpg)
RecallWe also interpret density functions as probabilities:
If p(x) is a probability density function (pdf), then
![Page 5: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/5.jpg)
Cumulative DistributionFunction
Suppose p(x) is a density function for a quantity.
The cumulative distribution function (cdf) for the quantity isdefined as
Gives:
• The proportion of population with value less than x
• The probability of having a value less than x.
![Page 6: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/6.jpg)
Example: A Spinner
Last class: A spinner that could take on any value0º ≤ x ≤ 360º.
Density function: p(x) = 1/360 if 0 ≤ x ≤ 360, and 0everywhere else.
![Page 7: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/7.jpg)
Example: A Spinner
Last class: A spinner that could take on any value0º ≤ x ≤ 360º.
Density function: p(x) = 1/360 if 0 ≤ x ≤ 360, and 0everywhere else.
CDF:
![Page 8: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/8.jpg)
Example: A Spinner
Density Function:Cumulative
Distribution Function:
![Page 9: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/9.jpg)
Properties of CDFs
• P(x) is the probability of values less than x– If P(x) is the cdf for the age in months of fish in
a lake, then P(10) is the probability a randomfish is 10 months or younger.
• P(x) goes to 0 as x gets smaller:
(In many cases, we may reach 0.)
![Page 10: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/10.jpg)
Properties of CDFs
• Conversely,
• P(x) is non-decreasing.– The derivative is a density function, which
cannot be negative.– Also, P(4) can’t be less than P(3), for example.
![Page 11: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/11.jpg)
PracticeLife expectancy (in days) of electronic component has densityfunction , for x ≥ 1, and p(x) = 0 for x < 1.
a) Probability component lasts between 0 and 1 day?
b) Probability it lasts between 0 and 10 days?
c) More than 10 days?
d) Find the CDF for the life expectancy.
![Page 12: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/12.jpg)
PracticeLife expectancy (in days) of electronic component has densityfunction , for x ≥ 1, and p(x) = 0 for x < 1.
a) Probability component lasts between 0 and 1 day?
Zero!
![Page 13: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/13.jpg)
PracticeLife expectancy (in days) of electronic component has densityfunction , for x ≥ 1, and p(x) = 0 for x < 1.
b) Probability it lasts between 0 and 10 days?
![Page 14: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/14.jpg)
PracticeLife expectancy (in days) of electronic component has densityfunction , for x ≥ 1, and p(x) = 0 for x < 1.
c) More than 10 days?
![Page 15: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/15.jpg)
PracticeLife expectancy (in days) of electronic component has densityfunction , for x ≥ 1, and p(x) = 0 for x < 1.
c) More than 10 days?
![Page 16: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/16.jpg)
PracticeLife expectancy (in days) of electronic component has densityfunction , for x ≥ 1, and p(x) = 0 for x < 1.
d) Find the CDF for life expectancy.
If x < 1, then
If x ≥ 1, then
![Page 17: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/17.jpg)
PracticeLife expectancy (in days) of electronic component has densityfunction , for x ≥ 1, and p(x) = 0 for x < 1.
d) Find the CDF for life expectancy.
![Page 18: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/18.jpg)
ExampleSomeone claims this is theCDF for grades on the 2015final exam.
Probability a random studentscored….
• 25 or lower?
1, or 100%.
• 50 or lower?
0.5, or 50%.
Conclusion?
They’re lying! This cannot bea cumulative distributionfunction! (It decreases.)
![Page 19: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/19.jpg)
Relating the CDF and DF
According the FTC version 2, since
then P'(x) = p(x).
,
![Page 20: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/20.jpg)
Relating the CDF and DF
According the FTC version 2, since
then P'(x) = p(x).
So the density function is the derivative, or rate of change,of the cumulative distribution function.
,
![Page 21: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/21.jpg)
ExampleSketch the density function forthe cdf shown.
![Page 22: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/22.jpg)
ExampleSketch the density function forthe cdf shown.
Slope?
![Page 23: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/23.jpg)
ExampleSketch the density function forthe cdf shown.
Slope?0
![Page 24: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/24.jpg)
ExampleSketch the density function forthe cdf shown.
Slope?0 Slope 1/2
![Page 25: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/25.jpg)
ExampleSketch the density function forthe cdf shown.
Slope?0 Slope 1/2
Slope 0
![Page 26: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/26.jpg)
ExampleSketch the density function forthe cdf shown.
Slope?0 Slope 1/2
Slope 0
Slope 1/2
![Page 27: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/27.jpg)
ExampleSketch the density function forthe cdf shown.
Slope?0 Slope 1/2
Slope 0
Slope 1/2
Slope 0
![Page 28: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/28.jpg)
ExampleSketch the density function forthe cdf shown.
Slope?0 Slope 1/2
Slope 0
Slope 1/2
Slope 0
DensityFunction
![Page 29: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/29.jpg)
DF vs. CDF
You must know which is which.
We work differently with density functions than withcumulative distribution functions.
![Page 30: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/30.jpg)
Another Example
Suppose the cumulative distribution function for the height oftrees in a forest (in feet) is given by
Find the height, x for which exactly half the trees are tallerthan x feet, and half the trees are shorter than x.
In case - Quadratic Formula:
�
x =!b ± b
2! 4ac
2a
![Page 31: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/31.jpg)
Another Example
Find the height, x for which exactly half the trees are tallerthan x feet, and half the trees are shorter than x.
�
0.1x ! 0.0025x 2= 0.5
!0.0025x 2+ 0.1x ! 0.5 = 0
!0.0025(x 2! 40x + 200) = 0
using quadratic : x = 20 ± 10 2 = 20 !10 2 " 5.86 ft
![Page 32: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/32.jpg)
![Page 33: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/33.jpg)
![Page 34: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/34.jpg)
(1)
(2)
(3)
(4)
ExampleAssume p(x) is the density function
p(x) =
1
2,!!!if !1! x ! 3
0,!!!if !x < 1!!or !!x > 3
"
#$
%$
Let P(x) represent the corresponding
Cumulative Distribution function.
Then (a) For the density function given above, what is the cumulative
distribution function P(x) for values of x between 1 and 3?
(b) P(5) = ?
(c) Choose the correct pair of graphs for p(x)!and!P(x) .
![Page 35: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/35.jpg)
In-class AssignmentThe time to conduct a routine maintenance check on a machine hasa cumulative distribution function P(t), which gives the fraction ofmaintenance checks completed in time less than or equal to tminutes. Values of P(t) are given in the table.
.98.80.38.21.08.030P(t), fraction completed302520151050t, minutes
a. What fraction of maintenance checks are completed in 15 minutesor less?
b. What fraction of maintenance checks take longer than 30minutes?
c. What fraction take between 10 and 15 minutes?
![Page 36: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/36.jpg)
In-class AssignmentThe time to conduct a routine maintenance check on a machine hasa cumulative distribution function P(t), which gives the fraction ofmaintenance checks completed in time less than or equal to tminutes. Values of P(t) are given in the table.
.98.80.38.21.08.030P(t), fraction completed302520151050t, minutes
a. What fraction of maintenance checks are completed in 15 minutesor less? 21%
b. What fraction of maintenance checks take longer than 30minutes? 2%
c. What fraction take between 10 and 15 minutes? 13%
![Page 37: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/37.jpg)
Mean and Median
Lesson 21
![Page 38: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/38.jpg)
The MedianThe median is the halfway point: Half thepopulation has a lower value, and half a highervalue.
If p(x) is a density function, then the median of thedistribution is the point M such that
p(x)dx!"
M
# = 0.5
![Page 39: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/39.jpg)
Suppose an insect life span ofmonths, and 0 elsewhere. What’s the median life span?
We need
Then solve for M:
months
Example
![Page 40: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/40.jpg)
Median From CDFIf we have a CDF P(x), the median M is the point where
P(M) = 0.5
(This of course says 1/2 of the population has a value lessthan M!)
![Page 41: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/41.jpg)
CDF: P(t) = t2, 0 ≤ t ≤ 1.
What’s the median of this distribution?
Example
![Page 42: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/42.jpg)
CDF: P(t) = t2, 0 ≤ t ≤ 1.
What’s the median of this distribution?
Example
![Page 43: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/43.jpg)
The Mean
What most people think of as the average:
Add up n values and divide by n.
How should we represent this for something represented by adensity function p(x)?
![Page 44: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/44.jpg)
The MeanWe define the mean of a distribution given by densityfunction p(x) to be
![Page 45: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/45.jpg)
Previous insect population example:
and p(x) = 0 elsewhere.
What’s the mean lifespan?
Example
![Page 46: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/46.jpg)
Previous insect population example:
and p(x) = 0 elsewhere.
What’s the mean lifespan? 8 months.
The mean is slightly below the median. This means
Example
![Page 47: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/47.jpg)
Previous insect population example:
and p(x) = 0 elsewhere.
What’s the mean lifespan? 8 months.
The mean is slightly below the median. This meansmore than half the insects live longer than the meanlifespan.
Example
![Page 48: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/48.jpg)
Example 2From “Another Example”,
�
p(x) = 0.1! 0.005x
mean = x p(x)dx!"
"
# = x(0
20
# 0.1! 0.005x)dx =
0.1x ! 0.005x2
0
20
# dx = 0.05x2 ! 0.0016 x
3
0
20
= 6.64 ft
![Page 49: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/49.jpg)
GeometryTurns out: the mean is the point where the graph of thedistribution would balance if we cut it out.
![Page 50: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/50.jpg)
Find the mean and median forand zero elsewhere.
Median: Solve for M
Example
![Page 51: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/51.jpg)
Find the mean and median forand zero elsewhere.
Mean: Evaluate
Example
![Page 52: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/52.jpg)
Find the mean and median forand zero elsewhere.
medianmean
Example
![Page 53: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/53.jpg)
A special distribution:
The Normal Distribution
![Page 54: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/54.jpg)
A special distribution:
The Normal Distribution
![Page 55: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/55.jpg)
Here’s an example with µ = 0 and σ = 1:
The distribution is symmetric about x = µ.
So the mean is µ, and so is the median.
σ is called the standard deviation, and describes how spreadout the curve is.
The Normal Distribution
![Page 56: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/56.jpg)
The Normal Distribution
Unfortunately, has no elementary antiderivative.
So we must use numerical means to evaluate integralsinvolving the normal distribution.
![Page 57: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/57.jpg)
Suppose a density function for number of people (inmillions) on the internet at one time were
What are the mean and median?
2
Example
![Page 58: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/58.jpg)
Suppose a density function for number of people (inmillions) on the internet at one time were
What are the mean and median?
Here, µ = 10, so both the mean and median are10 million people.
2
Example
![Page 59: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/59.jpg)
Suppose a density function for number of people (inmillions) on the internet at one time were
What is the standard deviation?
2
Example
![Page 60: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/60.jpg)
Suppose a density function for number of people (inmillions) on the internet at one time were
What is the standard deviation?
Here, we see that σ = 2, so 2 million people.
2
Example
![Page 61: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/61.jpg)
Suppose a density function for number of people (inmillions) on the internet at one time were
What is the standard deviation?
Here, we see that σ = 2, so 2 million people.
2
Example
![Page 62: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/62.jpg)
Suppose a density function for number of people (inmillions) on the internet at one time were
Write an integral for the fraction of time between 8and 12 million people are on the internet.
2
Example
![Page 63: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/63.jpg)
ExampleSuppose a density function for number of people (inmillions) on the internet at one time were
Write an integral for the fraction of time between 8and 12 million people are on the internet.
2
2
![Page 64: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/64.jpg)
How to calculate the required integral?
2
Evaluating the Integral
![Page 65: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/65.jpg)
How to calculate the required integral?
We could use our Simpson’s rule calculator:
2
Evaluating the Integral
![Page 66: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/66.jpg)
Evaluating the IntegralHow to calculate the required integral?
We could use our Simpson’s rule calculator:
2
n !x Result
2 2 0.693237
4 1 0.683058
8 0.5 0.682711
16 0.25 0.682691
32 0.125 0.68269
![Page 67: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/67.jpg)
How to calculate the required integral?
We could use our Simpson’s rule calculator:
2
n !x Result
2 2 0.693237
4 1 0.683058
8 0.5 0.682711
16 0.25 0.682691
32 0.125 0.68269
Seems to convergeto about 0.68.
Evaluating the Integral
![Page 68: Probability and Cumulative Distribution Functions · PDF fileProbability and Cumulative Distribution Functions Lesson 20. Recall ... If p(x) is a probability density function (pdf),](https://reader034.fdocuments.us/reader034/viewer/2022052313/5a715dc47f8b9ab1538cc236/html5/thumbnails/68.jpg)
In Class AssignmentFind the mean and median of the distribution withdensity function:
(calculations should be to 2 decimal places)
p(x) =1
4x
3 for 0 < x < 2!and!0!elsewhere