How is Poisson distribution related to Exponential distribution?

2 QUESTIONS PROBABILITY AND STATISTICS
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1. How is Poisson distribution related to Exponential distribution? In other words, if an event follows Poisson distribution what other even would have to follow the exponential distribution?

2. In excel, plot the PMF/PDFand CDF of Poisson, Exponential, and Normal distributions as a function of their parameters (Lambda for Poisson and Exponential – mean and variance for Normal).

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a. Determine initial values for X: To do this first generate equally spaced numbers between 0 and 10 (or more if you prefer).
b. Define initial values for parameters: Then use a separate cell with proper name to enter the initial value for parameters:
i. Lambda=1 for Poisson and Exponential(choose arbitrary initial values if you have more than one parameter; more than one parameter is possible depending on the formula you use).
ii. Mean=1 & variance=2 for Normal
c. Create three sheets: Copy the worksheet twice so you can plot each distribution separately. Name the resulting three worksheets according to the distribution you are plotting inside them.
d. Calculate using excel formulas: Then, in each of the three sheets, calculate the requested values(PDF, PMF, CDF) using excel formulas (you must use excel formulas, instead of calculating things outside).
e. Plot: Use a scatter plot (or other plotting methods that you prefer) to plot two-dimensional graphs for PDF, PMF, and CDFs. Give your graphs and its axes proper names. You should now have 6 graphs – two for each distribution.
f. Now make a copy of each worksheet, and increase the lambda value (first play with different values but ultimately set Lambda to 4). Explain how the distribution changes as lambda increases?

3. In semi-conductor manufacturing process, the maximum number of defects that we can accept on a wafer is 5. If the number of defects on a circuit follows a Poisson distribution with a mean of 2,
a. What is the probability that we reject a circuit? write the formula, write the value of each parameter (e.g. lambda), then provide the final value (no need to show the calculation steps; if you correctly write the value of parameters in the beginning you can only list the final value).

b. What percentage of circuits will be rejected in each lot?in each day? In each week?

c. On average, how many acceptable wafers do we make before we make an unacceptable wafer?

d. Imagine we can purchase a better machine that produces an average of 1 defect per wafer. Using the new machine, what percentage of each production batch will be unacceptable?

e. If the machine costs $1M and each unacceptable product costs us $200. At what level of production, would the investment in the new machine pay-off? How should the managers decide whether they should purchase the machine or not?

 

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