Probability Models
Assignment (4 Problems: 20 + 20+ 30 + 30 = 100 points total.)
Q Problem 1 An urn has 200 chips. Each chip has a color (clear, red, or green) and a size (large, medium, or small). The composition of chips in the urn is given in the table above (for example, there are 12 chips of green color and medium size). Suppose you choose the first chip uniformly at random from the urn.
•1.1 [5 points] What is the probability that the first chip is green?
•1.2 [5 points] What is the probability that the first chip is neither small nor green?
•1.3 [5 points] Given that the first chip is small, what is the probability that it is green?
•1.4 [5 points] Suppose the first chip is small and red. If you draw a second chip uniformly at random from the urn without replacing the first chip, what is the probability it is green?
Q Problem 2 Let X1, X2, X3 be Gaussian random variables each with µ = −2, σ2 = 9.
•2.1 [10 points] What is P (X3 < 0)?
•2.2 [10 points] Consider the random variable
X1 + X2 + X3
S3 = 3 .
What is the expected value E[S3] of the random variable S3?
Q Problem 3 Suppose a fair die with six faces “1, 2, 3, 4, 5, 6” is rolled 3 times with each roll independent of the others. Let X be the number of rolls resulting in the outcome “1”.
•3.1 [5 points] What is the probability that X = 0?
•3.2 [5 points] What is the probability that X = 3?
•3.3 [10 points] What is the probability density function (pdf) of X?
•3.4 [10 points] What is the expected value of X? Show your work.
Q Problem 4 Consider the continuous random variable X with probability density function
( ) = ( 1 (x3 + x) if 2 < x < 4
•4.1 [5 points] What is the probability that X = 3?
•4.2 [5 points] What is the probability that X < 3?
•4.3 [10 points] What is the expected value of X?
•4.4 [10 points] What is the variance of X?
Q Bonus [X points] Let X1, X2, . . . , XN be N standard Gaussian random variables (each of mean µ = 0 and variance σ2 = 1) and define
S = X1 + X2 + • • • + XN .N
The case N = 3 appears above in Problem 2.2. What is limN→∞ E[SN ] and why?
CS 340 Milestone One Guidelines and Rubric Overview: For this assignment, you will implement the fundamental operations of create, read, update,
Retail Transaction Programming Project Project Requirements: Develop a program to emulate a purchase transaction at a retail store. This
7COM1028 Secure Systems Programming Referral Coursework: Secure
Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip
CS 340 Final Project Guidelines and Rubric Overview The final project will encompass developing a web service using a software stack and impleme
