Use probability density functions (easily available online) equations to provide information about the theoretical range and number of parameters of the following distributions. Further, generate pdfs as column charts for these distributions by implementing the PDF in excel (and not using the
inbuilt excel functions). Vary the parameters to realize different shapes of the distributions.
1. Rayleigh
2. Poisson
3. Weibull
4. Log-normal
Distributions and Parameter Fitting
The following dataset provides the number of times chemical instruments of a particular batch
produced by a company experienced a technical glitch. It was found that 48 instruments did not
experience any technical glitch, 37 instruments experienced 1 technical glitch, and so on..
For this dataset,
1) Identify a probability distribution that best describes the data with correct reasoning
2) Using the steps described in the lecture, estimate the parameter(s) of the distribution that
best fits the data. Specifically,
a. Import the data into excel and indicate the name and probability distribution/mass
function (PDF/PMF) of the distribution you selected in Question 1 above.
b. Calculate the probabilities P(X = x) using the (PDF/PMF)
c. Calculate the log-likelihood by summing up the f_x*ln(P(X=x))
d. Use solver to Maximize the Log-Likelihood using appropriate constraints
e. Visualize the results
|
x
|
f_x
|
|
0
|
48
|
|
1
|
37
|
|
2
|
30
|
|
3
|
24
|
|
4
|
20
|
|
5
|
16
|
|
6
|
13
|
|
7
|
11
|
|
8
|
9
|
|
9
|
7
|
|
10
|
6
|
|
11
|
5
|
|
12
|
5
|
|
13
|
3
|
|
14
|
3
|
|
15
|
2
|
|
16
|
2
|
|
17
|
2
|
|
18
|
1
|
|
19
|
1
|
|
20
|
2
|
|
21
|
1
|
|
22
|
1
|
|
23
|
1
|
1. Highlight the vehicle fuel economy trends in the U.S. from 1975-2011
2. Highlight the trends in CO2 emission (in g/mi) in the U.S. due to vehicle emissions from 1975-
2011
3. Is there a correlation between the fuel economy and CO2 emissions? Explain your findings.
4. Using historical gas prices in the state of PA, ascertain the maximum and minimum gas prices
between 1983-2010.
5. Use curve fitting to appropriate subsets of the data to ascertain gas prices for 1975 to 1982 and
2011.
6. Using the average gas price per year, is there a correlation in PA gas prices and U.S. CO2
emissions between 1975-2011? Explain your findings.
7. Given your findings about vehicle fuel economy and CO2 emissions in US, propose one improvement to reduce CO2 emissions due to the transportation sector? In your answer, discuss the rationale and pros and cons of your suggested solution. Please cite your sources of inform.