Isye 6420

ISyE 6420 Homework 6 Cancer of Tongue $ 30.00 Buy This Answer; FINAL EXAM ISyE6420 Time to Second Birth Homework 2 ISyE 6420 Cell Clusters in 3D. [email protected] +1(541) 423-7793. Alabama.

Languages. Jupyter Notebook 99.1%. Other 0.9%. Contribute to woodyzc/ISYE6420 development by creating an account on GitHub.main. README. Stochastic price model simulation with CRR parameterization. Rodrigo Da Costa. Georgia Institute of Technology, Atlanta, GA 30332. Abstract. As part of …Solution Homework 2 ISyE 6420 February 15, 2021 We see that the above equation arrives at the kernel of beta distribution with parameters α = n + 1 and β = λ + ∑ n i =1 x i . Based on the problem statement, we know that n = 3, λ = 1 / 3 and ∑ 3 i =1 x i = 15. Therefore, the posterior distribution is gamma distribution G a (4, 46 / 3).Vineet Saini Final Project ISyE 6420 Dec 05, 2021 Problem Statement - Having worked in ecommerce domain as a software engineer, I have built enhancements to multiple platforms including logging various user statistics and activities they perform on ecommerce website. I have thought to give an analytical approach that can help company to analyse and predict the yearly spent by customers on ...How to prepare for ISYE 6420 Bayesian Statistics? The reviews on OMSCentral seem very mixed about the course. Some people say the course is good while others recommend taking the 6420 along side another coursera course to better explain the concepts.View Homework6solutions.pdf from CSC AI at Georgia State University. ISyE 6420 "Bayesian Statistics", Fall 2018 Homework 6 Solutions November 2018 1 Problem 1: Potato Leafhopper Consider the

Time-to-event Models: Gastric Cancer* — ISYE 6420 - BUGS to PyMC. import arviz as az import numpy as np import pymc as pm from pymc.math import exp %load_ext lab_black. 6. Time-to-event Models: Gastric Cancer* #. Adapted from code for Unit 8: gastric.odc. Data can be found here.Homework 5 ISyE 6420 Fall 2019 Due November 10, 2019, 11:55pm. HW5 is not time limited except the due date. Late submissions will not be accepted. Use of all available electronic and printed resources is allowed except direct com- munication that violates Georgia Tech Academic Integrity Rules.Sammy Gulrajani Homework 1 ISyE 6420 24-Jan-2020 2 Problem 2 To solve this problem, we first need to compute the conditional probability that the classifier will says the item was 1, given the item was actually 1 and the conditional probability that the classifier will say the item was 1, given the item was 0.Sep 11, 2022 · View ISYE - 6420_HW2 copy.docx from ISYE 6420 at Georgia Institute Of Technology. ISYE - 6420 Homework:2 Question 1. 2-D Density Task. Find (a) marginal distribution fX(x), (b) conditional ISyE 6420 1. Traffic. Marietta Traffic Authority is concerned about the repeated accidents at the intersection of Canton and Piedmont Roads. Bayes-inclined city-engineer would like to estimate the accident rate, even better, find a credible set.The Reserve Bank of India (RBI) is one of the few institutions Indians respect (paywall). India’s central bank is known for being clean, sensible, and well-run. It is also extremel...View Homework6solutions.pdf from CSC AI at Georgia State University. ISyE 6420 "Bayesian Statistics", Fall 2018 Homework 6 Solutions November 2018 1 Problem 1: Potato Leafhopper Consider theISYE 6420. Contribute to FengyiZhangcoding/ISYE-6420 development by creating an account on GitHub.

The price "messes up ones life," but it is also provides "greater opportunity" A Nobel prize “messes up one’s life a bit, because it comes with so many commitments—but it also give...View ISyE 6420 HW Template.docx from STA 653 at University of Kentucky. Homework 1 Erfu Yan ISyE 6420 August 20, 2019 Problem 1 Answer to the problem goes here. S1 = A4 A5 qS1 = 0.5 * 0.1 = 0.05 pS1Discover the best on-page SEO service provider in Jaipur. Browse our rankings to partner with award-winning experts that will bring your vision to life. Development Most Popular Em...© 2023 Georgia Institute of Technology video portal by KalturaKaltura

Emily matson died.

Homework 5 ISyE 6420 Spring 2020 Course Material for ISyE6420 by Brani Vidakovic is licensed under a Creative Commons Attribution- NonCommercial 4.0 International License. Due April 5, 2020, 11:55pm. HW5 is not time limited except the due date. Late submissions will not be accepted. Use of all available electronic and printed resources is allowed except direct com- munication that violates ...Rats Example with Missing Data* — ISYE 6420 - BUGS to PyMC. import arviz as az import numpy as np import pymc as pm from pymc.math import dot, stack, concatenate, exp, invlogit. 2. Rats Example with Missing Data* #. This example goes further into dealing with missing data in PyMC, including in the predictor variables.x5 = palatal width inside the second upper molars; x6 = width between the postglenoid foramina; x7 = interorbital width; x8 = least width of the braincase; x9 = crown length of the first upper molar. These 9 measurements (Columns 3-11) and gender (Column 2: male=0, female=1) are.View ISyE_6420_Midterm_Minhao_Leong.pdf from PROBABILIT at Caltech. Midterm Exam Minhao Leong ISyE 6420 March 1, 2020 Problem 1. Six Neurons. The following R-Code simulates 50,000,000 uniform randomView ISyE6420_HW1.pdf from ISYE 6420 at Georgia Institute Of Technology. Homework 1 Chen-Yang(Jim), Liu ISyE 6420 September 12, 2020 Problem 1 Answer to the problem goes here. E2 E1 E3 E4 E6 E7 E5 1.Final ISyE 6420 Problem 1 Answer to the problem goes here. a) Transformation of variables v and r to x1 and x2 is included in the WinBUGS code, see code in the appendix. b) Using the following setup and priors: logit ( P )=β 0 +β 1∗x 1+ β 2∗x 2 y B (1 , P. Solutions available.

An Overview of the Spacecraft - The Apollo spacecraft included many parts, which this overview of the spacecraft describes. Learn about the parts of the Apollo spacecraft in this o...View HW4.pdf from ISYE 6414 at Georgia Institute Of Technology. Homework 4 ISyE 6420 Fall 2020 1. Simple Metropolis: Normal Precision \u0010 \u0011 - Gamma. Suppose X = −2 was observed 1 from the populationSome life insurance policies are financial products that provide a death benefit to your beneficiaries. However, as you grow older, you may need skilled nursing care. If you lack s...ISyE 6420 Homework 6 solved 1. Cancer of Tongue. Sickle-Santanello et al (1988)1 provide data on 80 males diagnosed with cancer of the tongue. Data are provided in the file tongue.csv|dat|xlsx. The variables in the dataset are as follows: • Tumor DNA profile (1 - aneuploid tumor, 2 - diploid tumor); ...ISYE 6416 Computational Statistics; ISYE 6420 Bayesian Methods; ISYE 6664 Stochastic Optimization Free Electives (15 hours) Free electives may be any courses offered through the OMSCS program. You can find the list of current OMSCS courses here.View hw4.pdf from ISYE 6420 at Georgia Institute Of Technology. ISyE 6420 3/3/2021 Homework 4 Siyuan Li Problem 1 a) As the density function is = 0.6 × !"# # + 0.4 × candidate density is (0, 4!This is because PyMC uses more diagnostics to check if there are any problems with its exploration of the parameter space. Divergences indicate bias in the results. BUGS will happily run this model without reporting any problems, but it doesn’t mean that there aren’t any. For further reading, check out Diagnosing Biased Inference with ...View 6420HW3sol.pdf from ISYE 6420 at Georgia Institute of Cosmetology. 1 ISyE 6420 October 11, 2020 Homework 3 Solution Maxwell. (a) We let y = (y1 , . . . , yn ) and find the likelihood as n nAre ISYE 6402 and 6420 really so bad? Hey all -. These courses - time series analysis and Bayesian statistics - are some of the lowest rated classes on omscentral. The number of reviews and the consistency of the complaints makes me think that they probably are actually pretty bad, but I'm curious whether that tracks with the experiences ...

Everyone has a story about a nurse from Kerala. Whether you live in India or abroad, whether you’ve checked into a hospital as a patient or dropped in as a visitor, chances are you...

Using PyMC, pgmpy, NumPy, and other libraries to redo ISYE 6420: Bayesian Statistics at Georgia Tech in Python. The original course used Octave and OpenBUGS, and students have been requesting something more modern for years. Professor Vidakovic released his code under CC BY-NC 4.0, so I guess this is the same. Is that license meant for code? Detailed Course Information. Click the Schedule Type to find available offerings of the course on the Schedule of Classes. ISYE 6414 - Regression Analysis. Simple and multiple linear regression, inferences and diagnostics, stepwise regression and model selection, advanced regression methods, basic design and analysis of experiments, factorial ...This question is about the First Progress Platinum Prestige Mastercard® Secured Credit Card @CLoop • 03/15/22 This answer was first published on 03/15/22. For the most current info...Bayes' Theorem — ISYE 6420 - BUGS to PyMC. 4. Bayes' Theorem #. Say we have hypothesis H i and some data D. P ( H i ∣ D) = P ( D ∣ H i) P ( D) × P ( H i) We'll update the notation in Unit 4 when we start dealing with continuous distributions, but the structure won't change. P ( H i) represents the prior probability of H i.Add to Portfolio (opens a new window) ISYE 6620 - Discrete-Event Simulation. A thorough development of a simulation language is stressed in order to progress through a series of increasingly sophisticated applications of computer simulation. Projects cover a wide range of topics: production systems, inventory, finance, transportation, and ...This is because PyMC uses more diagnostics to check if there are any problems with its exploration of the parameter space. Divergences indicate bias in the results. BUGS will happily run this model without reporting any problems, but it doesn’t mean that there aren’t any. For further reading, check out Diagnosing Biased Inference with ...ISyE 6420 "Bayesian Statistics", Spring 2019 Midterm / Solutions March 23, 2019 1 Chad, Bayes, Car, and Vacation. We provide the following three approaches in solving the problem. (i) Use of WinBUGS/OpenBUGS The OpenBUGS code is shown below. The result by running the above code is (ii) Direct simulation using MATLAB. The MATLAB code for the simulation is shown below.Sep 11, 2022 · View ISYE - 6420_HW1 copy.docx from ISYE 6420 at Georgia Institute Of Technology. ISYE - 6420 Homework:1 Question1. Circuit. A circuit consisting of seven independent elements A1; : : : ;A7 is ISYE 6420 Homework 6. April 22, 2019 (This HW is for extra credit. You need to submit this HW only if you have lost some points in the previous HWs.) The data below shows the result of an experiment testing the insulation effect of a gas (SF 6 ). In the experiment a gaseous insulation was subjected to 100 high voltage pulses with a specified ...

Is there a qr code for zelle.

Promo code for oculus quest 2.

Can someone please let me know which one is better. I am deciding in taking ISYE 6414 or ISYE 6420? I love statistics and plan to pursue my PHD in statistics in Fall 2025 so my statistics background is actually advanced. I took two graduate statistics courses when I pursued my bachelor's degree in business so the concepts in these courses ...Repository of my ISYE6420 Bayesian Statistics coursework at GeorgiaTech: https://www2.isye.gatech.edu/~brani/isye6420 - abhiga/GeorgiaTech-ISYE6420-BayesianStatisticsHomework 5 ISyE 6420 Fall 2019 Due November 10, 2019, 11:55pm. HW5 is not time limited except the due date. Late submissions will not be accepted. Use of all available electronic and printed resources is allowed except direct com- munication that violates Georgia Tech Academic Integrity Rules.Metropolis example #. Let our model be a Gamma-Gamma conjugate model, where: X i | β ∼ Ga ( v, θ) θ ∼ Ga ( α, β) We’ll just have a single datapoint, x = 1, for simplicity. So if we let v = 1, α = 1, β = 1, our true posterior (see Conjugate table) will be G a ( 2, 2). We will use that to compare with our Metropolis results. For our ...I am taking ISYE 6420 Bayesian statistics course this semester and I want to understand my chances of surviving this course. I am stuck on Homework 3 to the point that every single thing feels like it is in Latin. I am seriously envisioning me not being able to turn anything in. I did absolutely fine on Homework 1 but Homework 2 I struggled a ...View 6420Midterm_sol.pdf from ISYE 6420 at Georgia Institute Of Technology. Midterm Solution 1 ISyE 6420 March 16, 2021 Problem 1. (a) As the engineer is 95% confidence that the probability θ ∈ [0.3,Course Syllabus: ISyE 6420 Bayesian Statistics 3 C 70-79% D 60-69% F 0-59% Description of Graded Components 1. There will be one midterm and one final exam that will be graded by faculty. The Midterm will be worth 25% of the course grade, while the Final will be worth 35% of the grade. 2.View Homework Help - Homework5.pdf from ISYE 6420 at Georgia Institute Of Technology. Homework 5 ISyE 6420 Brani Vidakovic 1. Prostate Cancer Data. This data set comes from the study by Stamey et alView ISYE6420_4.7_PRODUCTION_VERSION.pdf from ISYE 6420 at Georgia Institute Of Technology. Bayesian Statistics Empirical Bayes Brani Vidakovic Professor School of Industrial and Systems EngineeringHomework 2 ISyE 6420 Poisson distribution solved Sale! $ 35.00 $ 21.00 View This Answer; Homework 5 ISyE 6420 Bayesian Regression solved Sale! $ 35.00 $ 21.00 View This Answer; Homework 2 ISyE 6420 solved Sale! $ 35.00 $ 21.00 View This Answer; Homework 6 ISyE 6420 solved FINAL EXAM ISyE6420 Time to Second Birth solved ….

View MIDTERM EXAM.pdf from ISYE 6420 at Georgia Institute Of Technology. MIDTERM EXAM ISYE 6420 FALL 2020 Nb: see also the excel file for the detailed formulas. We want P(DA=1|BD=0, S=1), so weISyE 6420 1. Rinderpest Virus in Rabbits with Missing Data. Temperatures (temp) were recorded in a rabbit at various times (time) after the rabbit was inoculated with rinderpest virus (the data modified from Carter and Mitchell, 1958). Rinderpest (RP) is an infectious viral disease of cattle, domestic buffalo, and some species of wildlife; it ...Homework 5 ISyE 6420 Spring 2020 Course Material for ISyE6420 by Brani Vidakovic is licensed under a Creative Commons Attribution- NonCommercial 4.0 International License. Due April 5, 2020, 11:55pm. HW5 is not time limited except the due date. Late submissions will not be accepted.View Homework6_fall21.pdf from ISYE 6420 at Georgia Institute Of Technology. Homework 6 ISyE 6420: Fall 2021 1. Cancer of Tongue. Sickle-Santanello et al (1988)1 provide data on 80 malesView 6420HW6sol.pdf from ISYE 6420 at Georgia Institute Of Technology. ISyE 6420 "Bayesian Statistics", Spring 2019 Homework 6 / Solutions April 23, 2019 1 Squids with Missing Data. (a) We use theISYE 6669: Deterministic Optimization. Instructional Team. Santanu Dey Instructor. Kristina Gilmour Head TA. Overview. The course will teach basic concepts, models, and algorithms in linear optimization, integer optimization, and convex optimization. The first module of the course is a general overview of key concepts in optimization and ...A redo of ISYE 6420 code into Python \n. Using PyMC, pgmpy, NumPy, and other libraries to redo ISYE 6420: Bayesian Statistics at Georgia Tech in Python. The original course\nused Octave and OpenBUGS, and students have been requesting something more modern for years. \n. Professor Vidakovic released his code under CC BY-NC 4.0, so I …View ISYE6420 HW3.pdf from ISYE 6420 at Georgia Institute Of Technology. ISYE 6420 Feb 26, 2022 Homework 3 Allen Zhao Problem 1 Marietta Traffic Authority (a) Compare the Bayes estimator for with the Isye 6420, Wei Li Midterm ISyE 6420 February 29, 2020 Based on this credible set, 0 is included but it is close to the 2.5% percentile (-0.0377) so the likelihood of ࠵? 1-࠵? 2 = 0 is extremely low. The length of call between Hyla chrysoscelis and Hyla versicolor is mostly different and it is a special characteristic between the two., Homework 3 ISyE 6420 Spring 2020 Course Material for ISyE6420 by Brani Vidakovic is licensed under a Creative Commons Attribution- NonCommercial 4.0 International License. Due February 16, 2020, 11:55pm. HW3 is not time limited except the due date. Late submissions will not be accepted. Use of all available electronic and printed resources is allowed except direct com- munication that violates ..., Description. 1. Carpal Tunnel Syndrome Tests. Carpal tunnel syndrome is the most common. entrapment neuropathy. The cause of this syndrome is hard to determine, but it can include. trauma, repetitive maneuvers, certain diseases, and pregnancy. the nerve conduction velocity test. Tinel’s sign and Phalen’s test are both highly sensitive., 7. Ten Coin Flips Revisited: Beta Plots* — ISYE 6420 - BUGS to PyMC. import arviz as az import matplotlib.pyplot as plt import numpy as np import pymc as pm %load_ext lab_black. 7. Ten Coin Flips Revisited: Beta Plots* #. From Unit 4: betaplots.m. The professor goes back to the Ten Coin Flips example from Unit 2., Laplace's method is another integral approximation technique. This is faster than MCMC, but not as flexible. We expand the log of the function around its mode in a second-order Taylor expansion. This process results in a quadratic approximation of the function in the log space, which translates to a normal approximation in the original space., ISYE 6420 Homework 4 Solution, Spring 2019 Problem 1 (a) Laplace’s approximation Let γ = log(θ) denote the transformed parameter. Since the Laplace’s approximation uses normal density to approximate the target density, it is more appropriate to …, A Simple Regression* — ISYE 6420 - BUGS to PyMC. import arviz as az import matplotlib.pyplot as plt import numpy as np import pymc as pm %load_ext lab_black. 4. A Simple Regression* #. Adapted from unit 1: Regression.odc and unit 1: Regression.m. The professor shows an example of Bayesian linear regression in BUGS, and compares it to …, 7. Ten Coin Flips Revisited: Beta Plots* — ISYE 6420 - BUGS to PyMC. import arviz as az import matplotlib.pyplot as plt import numpy as np import pymc as pm %load_ext lab_black. 7. Ten Coin Flips Revisited: Beta Plots* #. From Unit 4: betaplots.m. The professor goes back to the Ten Coin Flips example from Unit 2., View Multinomial Regression — ISYE 6420 - BUGS to PyMC.pdf from ISYE 6420 at Georgia Institute Of Technology. Multinomial Regression Contents Problem statement import pymc as pm import numpy as, isye 6420 View More ISyE 6420 A/MSA: Bayesian Statistics Spring 2020 Instructor : Dr. Roshan Joseph 334 Groseclose Building Email: [email protected] , Phone: 404 894 0056 Time/Location : 10:10-11:00 AM on MWF in MRDC 2404 Office Hours : M 11:00 AM -12:00 PM or by appointment TA : Mr. Arvind Krishna, [email protected] Office hours: TBA., View ISYE - 6420_HW2 copy.docx from ISYE 6420 at Georgia Institute Of Technology. ISYE - 6420 Homework:2 Question 1. 2-D Density Task. Find (a) marginal distribution fX(x), (b) conditional, Metropolis example #. Let our model be a Gamma-Gamma conjugate model, where: X i | β ∼ Ga ( v, θ) θ ∼ Ga ( α, β) We'll just have a single datapoint, x = 1, for simplicity. So if we let v = 1, α = 1, β = 1, our true posterior (see Conjugate table) will be G a ( 2, 2). We will use that to compare with our Metropolis results. For our ..., Brozek index prediction* — ISYE 6420 - BUGS to PyMC. %load_ext watermark. 11. Brozek index prediction* #. This example goes over linear regression and Bayesian R 2. Adapted from Unit 7: fat1.odc, fat2d.odc, and fatmulti.odc. Data can be found here. Percentage of body fat, age, weight, height, and ten body circumference measurements (e.g ..., ISYE 6501 Introduction to Analytics (Easy course, weekly assignments, engaging crisp videos) ISYE 6669 Deterministic Optimization (Moderate course, weekly homework, exciting domain) ISYE 6420 Bayesian Statistics (Difficult course, biweekly homework, hard to grasp and follow) CSE 6242 Data and Visual Analytics (Moderate …, View HW4.pdf from ISYE 6414 at Georgia Institute Of Technology. Homework 4 ISyE 6420 Fall 2020 1. Simple Metropolis: Normal Precision \u0010 \u0011 - Gamma. Suppose X = −2 was observed 1 from the population, One of the requirements for creating a Facebook personal account, page or group is that you adhere to Facebook's name guidelines. These guidelines are designed to prevent people fr..., Four Local Area Dialogue Committees (LADCs) have been created following community elections in Ain Shams and Ezbet el-Nasr in Cairo, Markaz El-Abhath/ Warraq, Homework 2 ISyE 6420 Poisson distribution solved Sale! $ 35.00 $ 21.00 View This Answer; Homework 5 ISyE 6420 Bayesian Regression solved Sale! $ 35.00 $ 21.00 View This Answer; Homework 2 ISyE 6420 solved Sale! $ 35.00 $ 21.00 View This Answer; Homework 6 ISyE 6420 solved FINAL EXAM ISyE6420 Time to Second Birth solved, Brani Vidakovic / Greg Schreiter Exercises 4.8 ISyE 6420 Zeimet et al (2013) conducted a retrospective multicenter cohort study to determine expression of L1CAM by immunohistochemistry in 1021 endometrial cancer specimens with the goal to predict clinical outcome. Of 1021 included cancers, 17:7% were rated L1CAM-positive. Of these L1CAM-positive, ISyE 6420 January,27, 21. 1 Problem 1. Circuit. A circuit consisting of seven independent elements A 1 ,...,A 7 is connected as in Figure 1. Figure 1: Circuit S with seven independent elements The elements are operational during time interval T with probabilities. A 1 A 2 A 3 A 4 A 5 A 6 A 7 Probability of working (p) 0 0 0 0 0 0 0., The Yakovlev Yak-9 was instrumental in the Soviet Union's defeat of the Nazi Luftwaffe. Read about and see specifications of this classic airplane. Advertisement Yakovlev Yak-9 fig..., Add to Portfolio (opens a new window) ISYE 6620 - Discrete-Event Simulation. A thorough development of a simulation language is stressed in order to progress through a series of increasingly sophisticated applications of computer simulation. Projects cover a wide range of topics: production systems, inventory, finance, transportation, and ..., Basic Distributions — ISYE 6420 - BUGS to PyMC. 1. Basic Distributions # From this lecture, make sure you understand what a random variable is, the difference between …, IS 6420 - EXAM 2 QUIZZES & ANS!! IS 6420 - EXAM 2 QUIZZES & ANS!! 100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached. Previously searched by you. Previously searched by you. Sell. Where do you study. Your language. The Netherlands. United Kingdom. Germany. Spain. France. Belgium., Solution Homework 6 ISyE 6420 November 24, 2020 1 Potato Leafhopper. By running the OpenBUGS code (attached in Appendix A , we obtain the results shown in Figure Figure 1: OpenBUGS result for problem 1 (a) The estimated parameter β 0 and β 1 are 100.1 and -1.004, respectively., One fall class and one spring class for five years. This is due to heavy family commitments and work demands. Thank you in advance! — ISYE 6420 Bayesian Statistics also will be available for OMSCS students to take in Fall 2019. Again, the enrollment in this course will be limited., (RTTNews) - Cabot Oil & Gas Corp. (COG) said its Board has authorized an increase in the company's share repurchase program by 25.0 million shares... (RTTNews) - Cabot Oil & Gas Co..., View 6420HW5sol.pdf from ISYE 6420 at Georgia Institute Of Technology. Solution 1 Homework 5 ISyE 6420 April 6, 2021 Bipetal Diameter and Age of Fetus. We run OpenBUGS code and show the results in, © 2023 Georgia Institute of Technology video portal by KalturaKaltura, • The course project is an individual assignment. If you use part of someone else's work, you must include a citation for it. • Pick a dataset and perform some sort of Bayesian analysis on it. Ideas for data: data you collected or from your lab, from a published paper or from the Internet, preferably […], View ISYE - 6420_HW1 copy.docx from ISYE 6420 at Georgia Institute Of Technology. ISYE - 6420 Homework:1 Question1. Circuit. A circuit consisting of seven independent elements A1; : : : ;A7 is, {"payload":{"allShortcutsEnabled":false,"fileTree":{"":{"items":[{"name":"Q1","path":"Q1","contentType":"directory"},{"name":"Q3","path":"Q3","contentType":"directory ..., View ISyE6420_HW2.pdf from ISYE 6420 at Georgia Institute Of Technology. Homework 2 Chen-Yang(Jim), Liu ISyE 6420 September 20, 2020 Problem 1 Answer to the problem goes here. (a) marginal