- General probability concepts
What is probability density function (pdf)?
- What is probability mass function (pmf)?
- How is pdf different from pmf?
- What is cumulative density function (cdf)?
- How can CDF be computed indirectly from PMFs for a discreet event? Write the mathematical relationship.
- Normal distribution:
- Give an example of an event that can be expected to follow a Normal distribution?
- What are its mean and variance? Write the mathematical notation.
- What is its PDF? Write the formula and define each of the parameters.
- If we know the mean and variance of an event (e.g., X=x ; note: X is the variable which can take different values, x is the realization or the current value), how do we calculate the probability of X>0? What tables should we use and how do we use it?
- Poisson distribution:
- What quantity does it model?
- What is its PMF? Write the formula and define each of the parameters
- What is its mean? As a function of the parameters you used above.
- What is its variance? As a function of the parameters you used above.
- Exponential distribution:
- What quantity does it model?
- What is its PDF and CDF? Write the formula and define each of the parameters
- What is its CDF? Write the formula and define each of the parameters
- 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?
- 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).
- Determine initial values for X: To do this first generate equally spaced numbers between 0 and 10 (or more if you prefer).
- Define initial values for parameters: Then use a separate cell with proper name to enter the initial value for parameters:
- 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).
- Mean=1 & variance=2 for Normal
- 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.
- 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).
- 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.
- 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?
- 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,
- 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).
- What percentage of circuits will be rejected in each lot? in each day? In each week?
- On average, how many acceptable wafers do we make before we make an unacceptable wafer?
- 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?
- 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?
