Probability Concepts

Individual Assignment 01 – Probability Concepts

Due in 20 hours

SCM 432

Individual Assignment 01 – Probability Concepts

Name and ASU ID Number:

1.

General probability concepts:

a. What is probability density function (pdf)?

b. What is probability mass function (pmf)?

c. How is pdf different from pmf?

d. What is cumulative density function (cdf)?

e. How can CDF be computed indirectly from PMFs for a discreet event? Write the mathematical relationship.

2. Normal distribution:

a. Give an example of an event that can be expected to follow a Normal distribution?

b. What are its mean and variance? Write the mathematical notation.

c. What is its PDF? Write the formula and define each of the parameters.

d. 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?

3. Poisson distribution:

a. What quantity does it model?

b. What is its PMF? Write the formula and define each of the parameters

c. What is its mean? As a function of the parameters you used above.

d. What is its variance? As a function of the parameters you used above.

4. Exponential distribution:

a. What quantity does it model?

b. What is its PDF and CDF? Write the formula and define each of the parameters

c. What is its CDF? Write the formula and define each of the parameters

5. 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?

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

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?

7. 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?