Case Study > IE 322 Probabilistic Models in Industrial Engineering Case Study 1

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IE 322 Probabilistic Models in Industrial Engineering Case Study 1 Objective: This case study aims to apply the basic concepts of probability to the management of patient flows in an emergency department of Hospital X. Students are required to understand •

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the concepts of probability of events, conditional probability and independence, sampling and counting, and the fundamental rules including the additive rule and multiplicative rule. the concepts of bivariate distributions including joint probability mass function, marginal distributions and conditional distributions, as well as covariance and correlations.

Data Description: The data set contains 500 patient visits in the emergency department with the acuity level, the decision of admission (hospitalization) and the length of stay (ED Duration).

The acuity level is classified into five levels, L1-L5, with L1 being the least severity and L5 being the most. The admission decisions are labeled as 1 or 0, with 1 indicating that a patient was admitted (hospitalized) while 0 indicating that a patient was discharged. The date set is attached as an excel file.

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Part I: Basic Concepts of Probability 1. Provide a table of proportions of Admission/Discharge vs. Acuity Levels.

2. Plot the proportion of Admission Vs. Acuity Level (x-axis being the five acuity levels and y-axis being the proportion of patients being admitted given their acuity level). What can you tell from the plot?

3. What is the probability that a patient has a severity level L3?

4. What is the probability that a patient is admitted?

5. What is the probability that a patient has a severity level L3 and is admitted?

6. Is the event that a patient has a severity level of L3 and the event that a patient is admitted independent? Why or why not?

7. What is the probability that a patient is admitted given that she has a severity of L3?

What is the probability that a patient has a severity of L3 given that she is admitted?

8. What is the probability that a patient is admitted or has a severity level of L3? Can you also apply the additive rule to obtain this probability?

9. What is the probability that a patient is admitted given that she has an acuity level of at least L3? What is the probability that a patient is admitted and has an acuity level of at least L3?

10. Pick a random sample of 10 patients. (a) What is the probability that 5 patients will be discharged? (b) What is the probability that no more than 3 patients will be admitted? (c) What is the probability that no more than 1 patient is of acuity level L5? (d) What is the probability that no more than 1 patient is of at least the acuity level L4? (e) What is the probability that only 1 patient is of acuity level L3 and will be admitted? (f) What is the probability that 2 patients are of acuity level L5 and will be admitted?

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(g) What is the probability that at least 1 patient is of acuity level L5 and will be admitted?

Part II. Let X be the acuity level, taking values 1,2,3,4,5 and Y be the admission decision, taking values 1 or 0 with 1 indicating admission.

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1. What is the joint probability mass function of the acuity level X and admission decision Y? 2. What are the marginal distributions of the acuity level X and admission decision Y? 3. Calculate the probability that the acuity level is equal to 3 given that the person is admitted.

4. Calculate the probability that the acuity level is greater than or equal to 4 given that the person is admitted.

5. Calculate the probability that the person is admitted given that the acuity level is equal to 5.

6. Calculate the probability that the person is NOT admitted given that the acuity level is greater than or equal to 3.

7. Are the random variables representing the acuity level and admission decisions independent?

8. Given the joint probability mass function, calculate the means and variances of the acuity level and admission decision.

9. What is the covariance of the acuity level and admission decision?

10. What is the correlation of the acuity level and admission decision? What is the implication for the relationship of the acuity level and admission decision?

Part III. Let Z be the random variable indicating if the patient’s ED duration is longer than 330 minutes (5.5 hours). That is, Z=1 if the patient’s ED duration is longer than 330 minutes and Z=0 otherwise.

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1. Find the sample average/mean and variance of the 500 patients’ ED durations in the data.

2. What is the joint probability mass function of the acuity Level X and Z variable?

3. What is the joint probability mass function of the admission decision Y and Z variable?

4. What is the marginal distribution of the Z variable? What are the mean and variance of Z variable?

5. Calculate the probability that the acuity level is equal to three given that the ED Duration is more than 330 minutes, and the probability that the person is admitted given that the ED Duration is less than or equal to 330 minutes.

6. Calculate the probability that the ED Duration is less than or equal to 330 minutes given that the acuity level is equal to 3, and the probability that the ED Duration is more than 330 minutes given that the person is admitted.

7. Are the random variables representing the acuity level and ED Duration independent?

Are the random variables representing the admission decisions and the patient’s ED Duration independent?

8. What are the covariance and correlation of the patient’s acuity level and ED Duration?

What is the implication?

9. What are the covariance and correlation of admission decision and the patient’s ED Duration? What is the implication?

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