IHUMAN CASE STUDY TAMIKA BRAYTON REASON FOR ENCOUNTER FLANK PAIN 33 YEARS OLD, OUTPATIENT, Exams of Nursing

Download IHUMAN CASE STUDY TAMIKA BRAYTON REASON FOR ENCOUNTER FLANK PAIN 33 YEARS OLD, OUTPATIENT and more Exams Nursing in PDF only on Docsity!

IHUMAN CASE STUDY; TAMIKA BRAYTON

REASON FOR ENCOUNTER: FLANK PAIN 33

YEARS OLD, OUTPATIENT 2024

IHUMAN CASE STUDY TAMIKA BRAYTON

REASON FOR ENCOUNTER FLANK PAIN 33

YEARS OLD, OUTPATIENT 2024 ISYE 6644 -

FINAL PREP

1. History and Physical Examination

Purpose: To determine possible causes and help narrow differential diagnoses.

History: Ask about:

Onset, duration, and nature of the pain (sharp, dull, intermittent, or constant).

Associated symptoms: fever, hematuria, dysuria, nausea, vomiting, weight loss.

Past medical history: include all details of UTIs, kidney stones, or trauma.

Recent physical activity or injury.

Family history of kidney disease or stones.

Physical Exam:

Inspect: Note any bruising, skin changes, or obvious swelling.

Palpate: Evaluate tenderness over the costovertebral angle (CVA).

Percuss: CVA tenderness test to evaluate inflammation or stones of the kidneys.

Vital Signs: Check for fever, tachycardia, or hypertension.

2. Urinalysis (UA)

Purpose: To detect urinary abnormalities that suggest infection, stones, or

hematuria.

Procedure:

Obtain a clean-catch urine specimen.

Search for:

Microscopic hematuria: This indicates kidney stones or trauma.

Pyuria and bacteriuria: Suggests UTI or pyelonephritis.

Proteinuria: Might suggest glomerulonephritis or renal impairment.

Rating:

Positive nitrites/leukocyte esterase: UTI.

Crystals in urine: Indicates stones.

3. Blood Tests

Purpose: To assess renal function, infection, and systemic etiologies.

Tests and Rationale:

Complete Blood Count (CBC): Leukocytosis indicates infection (e.g.,

pyelonephritis).

Basic Metabolic Panel (BMP):

Elevated creatinine/BUN suggests renal dysfunction.

Hypercalcemia or hyperuricemia may suggest stone formation.

C-reactive Protein (CRP)/Erythrocyte Sedimentation Rate (ESR): High levels

indicate inflammation or infection.

Rating:

Interpret any abnormal renal or systemic markers to guide further testing.

4. Imaging Studies

Purpose: To visualize the urinary tract and identify anatomical causes.

Initial Imaging:

Ultrasound:

Non-invasive and helpful in detecting hydronephrosis, stones, or masses.

Preferred in pregnant patients or those with contraindications to radiation.

KUB X-ray

May detect radio-opaque stones but with low sensitivity.

Advanced Imaging:

CT Scan (Non-contrast):

Gold standard for kidney stone detection.

Identifies size, location, and complications such as obstruction.

MRI:

Used in patients with contraindications for CT contrast.

Assessment:

Stones: Management depends on size and location (eg, stones <5mm may pass

spontaneously).

Hydronephrosis: Indicates obstruction.

5. Urine Culture

Purpose: To confirm and identify the pathogen in suspected infections.

Procedure:

Get a midstream urine sample.

Culture to detect bacterial growth.

Test:

Guides antibiotic therapy in pyelonephritis or complicated UTIs.

6. Specialized Tests (if indicated)

Purpose: To help establish definite diagnosis when screening tests are not

definitive.

Cystoscopy: For hematuria or suspected bladder pathology.

24-Hour Urine Collection: Assesses metabolic etiologies for stones (e.g.,

hypercalciuria).

Renal Biopsy: For unexplained renal dysfunction or suspected glomerulonephritis.

7. Differential Diagnosis

Common Causes of Flank Pain:

Renal/Urinary: Kidney stones, pyelonephritis, hydronephrosis. Musculoskeletal:

Strain or trauma.

Other: Retroperitoneal masses, referred pain from gynecologic or gastrointestinal

issues. 8. Treatment Plan According to Test Results: Stones: Pain control,

hydration, possible urology referral. Infections: Antibiotics as per culture.

Obstruction: May need stenting or surgery. Non-urological Causes: Treat

underlying conditions (e.g., muscle relaxants for strain).

ISYE 6644 - Final prep 2024 What is a possible goal of an indifference-zone normal means selection technique? - CORRECT ANSWER>>>>Find the normal population having the largest mean, especially if the largest mean is ≫ the second-largest. TRUE or FALSE? The Bechhofer procedure for selecting the normal population with the largest mean specifies the appropriate number of observations to take from each competing population, and simply selects the competitor having the largest sample mean. - CORRECT ANSWER>>>>True TRUE or FALSE? Sometimes a single-stage procedure like Bechhofer's is inefficient. In fact, it's possible to use certain sequential procedures that take observations one-at-a-time (instead of all at

once in a single stage) to make good selection decisions using fewer observations. - CORRECT ANSWER>>>>True For which scenarios(s) below might it be appropriate to use a Bernoulli selection procedure? a) Find the inventory policy having the largest profit. b) Find the drug giving the best chance of a cure. c) Find the maintenance policy having the lowest failure probability. d) Find the scheduling rule that that has the best chance of making an on-time delivery. - CORRECT ANSWER>>>>All three of (b), (c), and (d). Suppose that a Bernoulli selection procedure tells you to take 100 observations from each of two populations, A and B. It turns out that A gets 85 successes and B gets 46 successes. What do you think? - CORRECT ANSWER>>>>1) A almost certainly has a higher success probability than B.

2. We could've probably stopped sampling a bit earlier (i.e., with fewer than 100 observations) because A was so far ahead of B. For which scenarios(s) below might it be appropriate to use a multinomial selection procedure? - CORRECT ANSWER>>>>Find the most-popular political candidate. Suppose that we want to know which of Coke, Pepsi and Dr.pepper is the most popular. We would like to make the correct selection with probability of at least P=0.90 in the event that the ration of the highest-to-second-highest preference probabilities happens to be at least 0=1.4. How many people does the single-stage procedure Mbem require us to interview? - CORRECT ANSWER>>>> 126 Go to the table in the notes and pick off the entry for k=3, P^⋆=0.90, and θ^⋆=1.4. Which of the following parameters can you get confidence intervals for? Means Variances Quantiles Differences between the means of two systems - CORRECT ANSWER>>>>All of the above If we have an iid normal sample of observations, X1, X2,...Xn, what probability distribution is most- commonly used to obtain cofidence intervals for the mean? - CORRECT ANSWER>>>>t TRUE or FALSE? The paired CI for the differences in two means is designed to work especially well if all of the observations from the first population are completely independent of all of the observations from the second population. - CORRECT ANSWER>>>>FALSE. {In fact, it's easier to distinguish between the two means if Xi is positively correlated with Yi. Think about my parallel parking example in the class notes.} TRUE or FALSE? You can use a version of independent replications to obtain confidence intervals for the difference in the means from two simulation models. - CORRECT ANSWER>>>>TRUE TRUE or FALSE? The common random numbers technique intentionally induces positive correlation between two systems - much like a paired-𝑡confidence interval. - CORRECT ANSWER>>>>TRUE CRN depends on someone's ability to manipulate the underlying pseudo-random numbers - e.g., use the same arrival times when simulating two competing simulated systems. So who ultimately controls those PRNs?? - CORRECT ANSWER>>>>You do - you are powerful!

TRUE or FALSE? The antithetic random numbers technique intentionally induces negative correlation between two runs of the same system - this allows us to better estimate the mean of the system. - CORRECT ANSWER>>>>TRUE TRUE or FALSE? The control variates technique provides unbiased, low-variance estimates using a method reminiscent of regression. - CORRECT ANSWER>>>>TRUE Statistical ranking and selection techniques have been designed to address a variety of comparison problems. Which ones from the following list? Find the population having the largest mean. Find the system with the smallest variance. Find the alternative with the highest success probability. Find the most-popular candidate. All of the above. - CORRECT ANSWER>>>>All of the above. Suppose we are dealing with i.i.d. normal observations with unknown variance. Which of the following is true about a 95% confidence interval for the mean μ? - CORRECT ANSWER>>>>We are 95% sure that our CI will actually contain the unknown value of μ. We are studying the waiting times arising from two queueing systems. Suppose we make 4 independent replications of both systems, where the systems are simulated independently of each other. replication system 1 system 1 10 25 2 20 10 3 5 40 4 30 30 Assuming that the average waiting time results from each replication are approximately normal, find a two-sided 95% CI for the difference in the means of the two systems. - CORRECT ANSWER>>>>This is a two-sample CI problem assuming unknown and unequal variances. We have [-29.76, 9.76] This is sort of the same as Question 2, except we have now used common random numbers to induce positive correlation between the results of the two systems. Again find a two-sided 95% CI for the difference in the means of the two systems. - CORRECT ANSWER>>>>This is a paired-t CI problem assuming unknown variance of the differences. [-16.5, - 3.5] Suppose A and B are two identically distributed, unbiased, antithetic estimators for the mean μ of some random variable, and let C = ( A + B ) / 2. Which of the following is true? - CORRECT ANSWER>>>>E [ C ] = μ and V a r ( C ) < V a r ( A ) / 2. Suppose that you want to pick that one of three normal populations having the largest mean. We'll assume that the variances of the three competitors are all known to be equal to σ 2 = 4. (Ya, I know that this is a crazy, unrealistic assumption, but let's go with it anyway, okey dokey?) I want to choose the best of the three populations with probability of correct selection of 95% whenever the best population's mean happens to be at least δ ⋆ = 1 larger than the second-best population's. How many observations from each population does Bechhofer's procedure N B tell me to take before I can make such a conclusion? - CORRECT ANSWER>>>>Using the notation of the notes, we want to make sure to get the right answer with probability of P ⋆ = 0.95 whenever μ [ k ] − μ [ k − 1 ] ≥ δ ⋆ = 1.

We simply go to NB's table with k = 3 and δ ⋆ / σ = 1 / 2 to obtain a sample size of n = 30 from each population. In the above problem, suppose that we take the necessary observations and we come up with the following sample means: X ¯ 1 = 7.6, X ¯ 2 = 11.1, and X ¯ 3 = 3.6. What do we do? - CORRECT ANSWER>>>>Pick population 2 and say that we are right with probability at least 95% Suppose that we want to know which of Coke, Pepsi, and Dr. Pepper is the most popular. We would like to make the correct selection with probability of at least P ⋆ = 0.90 in the event that the ratio of the highest-to-second-highest preference probabilities happens to be at least θ ⋆ = 1.4. If we use procedure M B E M, then the corresponding table in the notes (with k = 3) tells us to take 126 samples (taste tests). Suppose we take those samples sequentially and after 100 have been taken it turns out that 65 people prefer Coke, 25 love Pepsi, and 10 like Dr. Pepper. What to do? - CORRECT ANSWER>>>>Stop the test now and declare with confidence of at least 90% that Coke is the most-preferred. Which of the following problems might best be characterized by a finite-horizon simulation? - CORRECT ANSWER>>>>Simulating the operations of a bank from 9:00 a.m. until 5:00 p.m. Let's run a simulation whose output is a sequence of daily inventory levels for a particular product. Which of the following statements is true? - CORRECT ANSWER>>>>The consecutive daily inventory levels may not be identically distributed. Suppose that X 1 , X 2 , ... is a stationary (steady-state) stochastic process with covariance function R k ≡ C o v ( X 1 , X 1 + k ), for k = 0 , 1 , .... We know from class that the variance of the sample mean can be represented asV a r ( X ¯ n ) = 1 n [ R 0 + 2 ∑ k = 1 n − 1 ( 1 − k n ) R k ] .We also know from class that for a simple AR(1) process, we have R k = ϕ k, k = 0 , 1 , 2 , ... Compute V a r ( X ¯ n ) for an AR(1) process with n = 3 and ϕ = 0.8. - CORRECT ANSWER>>>>0. Suppose we want to estimate the expected average waiting time for the first m = 100 customers at a bank. We make r = 4 independent replications of the system, each initialized empty and idle and consisting of 100 waiting times. The resulting replicate means are: i 1 2 3 4 Z i 5.2 4.3 3.1 4. Find a 90% confidence interval for the mean average waiting time for the first 100 customers. - CORRECT ANSWER>>>>[3.188,5.212] Consider a particular data set of 100,000 stationary waiting times obtained from a large queueing system. Suppose your goal is to get a confidence interval for the unknown mean. Would you rather use (a) 50 batches of 2000 observations or (b) 10000 batches of 10 observations each? - CORRECT ANSWER>>>>50 batches of 2000 observations because the method of batch means requires a very large batch size Consider the output analysis method of non overlapping batch means. Assuming that you have a sufficiently large batch size, it can be shown that when the number of batches b is even, the expected width of the 90% two-sided confidence interval for μ is proportional tot 0.05 , b − 1 b − 1 ( b − 1 2 ) ( b − 3 2 ) ⋯ 1 2 ( b − 2 2 )! .Using the above equation, determine which of the following values of b gives the smallest expected width. - CORRECT ANSWER>>>>b= Let h ( b ) denote the value of the above expression as a function of b. Then easy calculations reveal that h ( b ) = 3.157, h ( 4 ) = 1.019, and h ( 6 ) = 0.845. So the answer is b = 6 Consider the following observations: 54 70 75 62 If we choose a batch size of 3, calculate all of the overlapping batch means for me. - CORRECT ANSWER>>>>66.3, 69.

X1,3 = 1/3 ΣXi = 66.3 and X2,3 = 1/3 ΣXi = 69. TRUE or FALSE? Simulation output (e.g., consecutive customer waiting times) is almost never i.i.d. normal - and that's a big fat problem - CORRECT ANSWER>>>>TRUE We often distinguish between two general types of simulations with regard to output analysis. What are they called? - CORRECT ANSWER>>>>Finite-horizon and steady-state TRUE or FALSE? Suppose that X 1 , X 2 ,... , X n are consecutive waiting times, and we define the sample mean X ¯ = ∑ i = 1 n X i / n. Then V a r ( X ¯ ) = V a r ( X i ) / n. - CORRECT ANSWER>>>>Very FALSE! (The issue is that correlation between the observations messes up the variance of the sample mean. In fact, this is one of the main reasons why output analysis is difficult!) Which of the following scenarios might be well-suited for a finite-horizon analysis? - CORRECT ANSWER>>>>Simulate bank operations from 8:00 a.m. to 5:00 p.m. Simulate an inventory system until the first stock-out occurs TRUE or FALSE? The main method of attack for terminating simulations is via independent replications. - CORRECT ANSWER>>>>TRUE (Independent replications help to mitigate issues with simulation outputs not being i.i.d.) TRUE or FALSE? You can also conduct finite-horizon estimation for quantities other than expected values, e.g., simulate a bank from 8:00 a.m. to 5:00 p.m., and find a confidence interval for the 95th quantile of customer waiting times. - CORRECT ANSWER>>>>TRUE (the method of independent replications allows us to determine sample variances, which subsequently allows us to determine confidence intervals) How can we deal with initialization bias if we want to do a steady-state analysis? - CORRECT ANSWER>>>>Make an extremely long run in order to overwhelm it. Truncate (delete) some of the initial data. Which of the following scenarios might be well-suited for a steady-state analysis? - CORRECT ANSWER>>>>Simulate an assembly line working 24/ A Markov chain simulated until the transition probabilities appear to converge Which of the following is useful for analyzing steady-state simulation output? - CORRECT ANSWER>>>>The method of batch means Which of the following statements is true? a) The method of batch means is easy to use. b) Batch means chops the consecutive observations into a number of nonoverlapping, contiguous batches c) You can use the method of batch means to obtain a confidence interval for the steady-state mean 𝜇. d) The batch means estimator for the variance parameter 𝜎^2 is asymptotically unbiased as the batch size 𝑚→∞. e) All of the above - CORRECT ANSWER>>>>All of the above Which of the following methods can be used for steady-state analysis? a. Batch Means b. Independent Replications (though it might suffer some minor(?) initialization problems) c. Overlapping Batch Means (something for nothing!)

d. Regeneration e. Standardized Time Series f. All of the above - CORRECT ANSWER>>>>f. All of the above It's GIGO time! Let's consider an M / M / 1 queueing system with Exp(λ) interarrivals and Exp(μ) FIFO services at a single server. You may recall from some class (either this one or stochastic processes) that the steady-state expected cycle time (i.e., the time that the customer is in the system, including wait + service) is w = 1 / ( μ − λ ). If you were to try this out in Arena, let's say with E X P O ( 10 = 1 / λ ) interarrivals and E X P O ( 8 = 1 / μ ) services (note the notation change between my usual "Exp" and Arena's " E X P O "), then we'd get w = 1 / ( 0.125 − 0.1 ) = 40. Go ahead, see for yourself in Arena, but make sure that you run the system for 100,000 or so customers so that you can be sure that you're in steady-state! Finally, here's the GIGO question, which will show what can happen when you mis-model a component of your process: What is the (approximate) steady-state expected cycle time if you hav - CORRECT ANSWER>>>>about 23 The U N I F case has waaaay smaller tails than the E X P O, so it's reasonable to assume that the cycle times will tend to be lower for the U N I F case. In fact, after 100,000 customers in Arena, I got an average time of 23.5. Let's play Name That Distribution! The number of times a "3" comes up in 10 dice tosses. - CORRECT ANSWER>>>>Binomial Name That Distribution! The number of dice tosses until a 3 comes up. - CORRECT ANSWER>>>>Geometric Name That Distribution! The number of dice tosses until a 3 comes up for the 4th time. - CORRECT ANSWER>>>>Negative Binomial Name That Distribution! IQs - CORRECT ANSWER>>>>Normal Name That Distribution! Cases in which you have limited information, e.g., you only know the min, max, and "most likely" values that a random variable can take. - CORRECT ANSWER>>>>Triangular Find the sample variance of - 3, - 2, - 1, 0,1,2,3 - CORRECT ANSWER>>>>14/ S^2 formula If X 1 , ... , X 10 are i.i.d. Pois(6), what is the expected value of the sample variance S^2? - CORRECT ANSWER>>>> S^2 is always unbiased for the variance of X i. Thus, we have E [ S^2 ] = V a r ( X i ) = λ = 6 Suppose that estimator A has bias = 3 and variance = 12, while estimator B has bias - 2 and variance = 14. Which estimator (A or B) has the lower mean squared error? - CORRECT ANSWER>>>>MSE = Bias^ 2 + Var, so M S E ( A ) = 9 + 12 = 21 and M S E ( B ) = 4 + 14 = 18. If X 1 = 2, X 2 = − 2, and X 3 = 0 are i.i.d. realizations from a Nor(μ , σ^2) distribution, what is the value of the maximum likelihood estimate for the variance σ^2? - CORRECT ANSWER>>>>σ^2 = (n-1)/n *S^ 8/

Suppose we observe the Pois( λ ) realizationsX 1 = 5 , X 2 = 9 and X 3 = 1. What is the maximum likelihood estimate of λ? - CORRECT ANSWER>>>>The likelihood function is L ( λ ) Thus, ln ( L ( λ ) ) This implies that (d /d λ )*ln ( L ( λ ) ) = − n + (∑ i = 1 n x i )/ λ. Setting the derivative to 0 and solving yields λ ^ = x ¯ = 5. Duh! What a surprise! Suppose that we have a number of observations from a Pois( λ ) distribution, and it turns out that the MLE for λ is λ ^ = 5. What's the maximum likelihood estimate of Pr ( X = 3 )? - CORRECT ANSWER>>>>By invariance, Pr ^ ( X = k ) = e − λ ^ λ ^ k k! , so that we have Pr ^ ( X = 3 ) = e − x ¯ x ¯ k k! = e − 5 5 3 3! = 0.1404. Suppose that we observe X 1 = 5, X 2 = 9, and X 3 = 1. What's the method of moments estimate of E [ X 2 ]? - CORRECT ANSWER>>>>The MOM estimator for E[X^2] = ( 25 + 81 + 1 ) / 3 = 107 / 3 Consider the PRN's U 1 = 0.1 , U 2 = 0.9 , and U 3 = 0.2. Use Kolmogorov-Smirnov with α = 0.05 to test to see if these numbers are indeed uniform. Do we ACCEPT or REJECT uniformity? - CORRECT ANSWER>>>>Accept Since D < D α , n, we ACCEPT uniformity (though it's kind of a joke since it's only based on 3 observations) What does GIGO mean? - CORRECT ANSWER>>>>Garbage-in-garbage-out What's a good distribution for modeling heights of people? - CORRECT ANSWER>>>>Normal What's a good distribution for modeling the number of random customer arrivals to a store? - CORRECT ANSWER>>>>The Poisson distribution is used for counting the number of arrivals over some interval of time. TRUE or FALSE? If the expected value of your estimator equals the parameter that you're trying to estimate, then your estimator is unbiased. - CORRECT ANSWER>>>>TRUE. (This is the definition of unbiasedness in words.) TRUE or FALSE? If X 1 , X 2 ,... , X n are i.i.d. with mean μ, then the sample mean X ¯ is unbiased for μ. - CORRECT ANSWER>>>>TRUE What is the MSE of an estimator? - CORRECT ANSWER>>>>𝐵𝑖𝑎𝑠^2+𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Suppose that X 1 = 4 , X 2 = 3 , X 3 = 5 are i.i.d. realizations from an E x p ( λ ) distribution. What is the MLE of λ? - CORRECT ANSWER>>>>Since λ^ = 1/ X ¯ = 0. TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the N o r ( μ , σ 2 ) distribution. - CORRECT ANSWER>>>>TRUE (it's possible based on taking the partial derivatives of the likelihood function with respect to each parameter). TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. - CORRECT ANSWER>>>>TRUE. (Think of the gamma example that we did.)

Suppose that the MLE for a parameter θ is θ ^ = 4. Find the MLE for θ. - CORRECT ANSWER>>>>Invariance immediately implies that the MLE of sqrt(θ) is simply sqrt( θ ^) = 2θ ^ = 2 If you're in this course, then you should always love your... - CORRECT ANSWER>>>>Method of Moments! TRUE or FALSE? If X 1 , X 2 ,... , X n are i.i.d., then the MoM estimator for E [ X 3 ]is 1 n ∑ i = 1 n X i 3. - CORRECT ANSWER>>>>TRUE (by definition of MoM estimator) Suppose H 0 is true, but you've just rejected it! What have you done? - CORRECT ANSWER>>>>You've committed a Type I error. Suppose that you are testing 100 observations to see if they are exponential (with unknown rate parameter λ). You decide to break the hypothesized p.d.f. into 5 intervals. How many degrees of freedom will your resulting chi-square goodness-of-fit test statistic have? - CORRECT ANSWER>>>>The degrees of freedom is k−1−s=5−1−1= TRUE or FALSE? The Weibull distribution is a special case of the exponential distribution - CORRECT ANSWER>>>>FALSE. (It's the other way around!) TRUE or FALSE? A search algorithm such as bisection or Newton is required to obtain the MLEs for the two Weibull parameters. - CORRECT ANSWER>>>>TRUE (the r parameter of the Weibull distribution cannot be determined in closed form). goodness-of-fit tests - CORRECT ANSWER>>>>Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling Shapiro-Wilk Which of the following problematic issues can arise in input data analysis? - CORRECT ANSWER>>>>Not enough data Data coming from strange-looking, "non-standard" distributions Nonstationary data (in which the distribution appears to change over time) Correlated data TRUE or FALSE? Arena has an Input Analyzer that automates distribution fitting for certain well- known distributions. - CORRECT ANSWER>>>>TRUE. (It's often a great time-saver! Suppose that a machine consists of components A, B, and C. If any of the components fail, then the machine breaks down. The failure times for A, B, and C are exponential (with probability 0.5), normal (w.p. 0.1), and Weibull (w.p. 0.4), respectively. YES or NO? Would the composition method be a good choice to generate the overall machine breakdown time for this scenario? - CORRECT ANSWER>>>>YES. Composition would be much more efficient than generating all three failure times for A, B, and C, and then taking the minimum. Let H and W denote a person's height and weight. Which of the following best describes the joint distribution of ( H , W )? - CORRECT ANSWER>>>>Bivariate normal distribution with positive correlation coefficient The multivariate normal distribution's covariance matrix ∑ can be decomposed into the form ∑ = C C T. What does the C informally stand for? - CORRECT ANSWER>>>>Cholesky TRUE or FALSE? The times between Poisson( λ ) arrivals are i.i.d. Exp( λ ). - CORRECT ANSWER>>>>TRUE. (... as long as ( λ ) doesn't change over time.)

TRUE or FALSE? The times between nonhomogeneous Poisson( λ ) arrivals are i.i.d. Exp( λ ). - CORRECT ANSWER>>>>FALSE. (They're independent, but the distribution changes as the arrival rate λ ( t ) changes over time.) What is the recommended method to generate nonhomogeneous Poisson arrivals? - CORRECT ANSWER>>>>Thinning Consider the time series Y i = ϕ Y i − 1 + ϵ i , i = 1 , 2 , ..., where the ϵ i's are i.i.d. normal. What is this time series called? (Give the best answer.) - CORRECT ANSWER>>>>first-order autoregressive process Consider the waiting time W i Q of the i th customer in an M / M / 1 queuing system with first-in-first- out services. What relation allows you to calculate customer ( i + 1 )'s waiting time W i + 1 Q based on W i Q, customer i's service time S i, and customer ( i + 1 )'s interarrival time I i + 1? - CORRECT ANSWER>>>>Lindley's equation W^Q_{i+1} = max{0, W^Q_i + S_i - I_{i+1}} Brownian motion is also known as ... - CORRECT ANSWER>>>>Wiener process Let W ( t ) denote a Brownian motion process at time t. What is C o v ( W ( 2 ) , W ( 3 ) )? - CORRECT ANSWER>>>>min(s, t) 2 TRUE or FALSE? Unif(0,1) pseudo-random numbers can be used to generate pretty much any other random variates, e.g., exponential, normal, and Poisson. - CORRECT ANSWER>>>>TRUE. That's the point of this module! If 𝑋 is a continuous random variable with c.d.f. 𝐹(𝑥), what's the distribution of 𝐹(𝑋)? - CORRECT ANSWER>>>>Unif(0,1) If 𝑈 is a Unif(0,1) random variable, what's the distribution of −1/𝜆𝑙𝑛(1−𝑈)? - CORRECT ANSWER>>>>Exp( 𝜆 ) f 𝑈 is a Unif(0,1) random variable, what's the distribution of 1/3[−ln(U)]^1/2? - CORRECT ANSWER>>>>Weibull, with parameters λ=3 and β= TRUE or FALSE? You can find the inverse c.d.f. Φ^−1(. ) of the standard normal distribution in closed form. - CORRECT ANSWER>>>>FALSE. You need to use an approximation. If 𝑈 is Unif(0,1), what is ⌈ 6 𝑈⌉? (Note that ⌈.⌉ is the round-up function.) - CORRECT ANSWER>>>>A 6 - side die toss If 𝑈 is Unif(0,1), what is ⌈𝑙𝑛(𝑈)/𝑙𝑛(5/6)⌉? (Note that ⌈.⌉ is the round-up function.) - CORRECT ANSWER>>>>Geom(1/6) TRUE or FALSE? If you can't find a good theoretical distribution to model a certain random variable, you might want to use the empirical distribution of the data to do so. - CORRECT ANSWER>>>>TRUE. That's the point of this lesson! TRUE or FALSE? The convolution method involves sums of random variables. - CORRECT ANSWER>>>>TRUE

Support that U1 and U2 are PRNs. What's the distribution of U1 + U2? - CORRECT ANSWER>>>>Triangular (0,1,2) Suppose that I want to generate a simple Unif(2/3, 1) via A-R. Support I generate a PRN U1=0.16. Do I accept U1 as my Unif(2/3, 1)? - CORRECT ANSWER>>>>NO. In this example, we only accept if U1 ≥ 2/3; so we reject and try again until we meet that condition. TRUE or FALSE? The proof that A-R works is really easy. - CORRECT ANSWER>>>>FALSE. Super false, in fact! Suppose that 𝑋 is a continuous RV with p.d.f. 𝑓(𝑥)=30𝑥^4(1−𝑥), for 0<𝑥<1. Why is acceptance- rejection a good method to use to generate 𝑋? - CORRECT ANSWER>>>>Because the c.d.f. of 𝑋 is very hard to invert True or False? The A-R algorithm for X~Pois(λ) tells us to generate PRNs until e^-λ .... Ui for the first time, and then set X=n. - CORRECT ANSWER>>>>TRUE. Yup - that's how you do it (though sometimes it's a bit tedious if λ is large). Unif(0,1) PRNs can be used to generate which of the following random entities? - CORRECT ANSWER>>>>All of the above --- and just about anything else! If X is an Exp(λ) random variable with c.d.f. F ( x ) = 1 − e − λ x, what's the distribution of the random variable 1 − e − λ X? - CORRECT ANSWER>>>>Unif(0,1) If U is a Unif(0,1) random variable, what's the distribution of − 1 λ ln ( U )? - CORRECT ANSWER>>>>Exp(λ) Suppose that U 1 , U 2 , ... , U 5000 are i.i.d. Unif(0,1) random variables. Using Excel (or your favorite programming language), simulate X i = − ln ( U i ) for i = 1 , 2 , ... , 5000. Draw a histogram of the 5000 numbers. What p.d.f. does the histogram look like? - CORRECT ANSWER>>>>Exponential Suppose the c.d.f. of X is F ( x ) = x^3 / 8, 0 ≤ x ≤ 2. Develop a generator for X and demonstrate with U = 0.54. - CORRECT ANSWER>>>>X = 2 U^{1/3} = 1. If X is a Nor(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f., what is the distribution of Φ ( X )?

• CORRECT ANSWER>>>>Uniform If U is a Unif(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f., what is the distribution of 2 Φ − 1 ( U ) + 3? - CORRECT ANSWER>>>>Nor(3,4) How would you simulate the sum of two 6-sided dice tosses? (Note that ⌈ ⋅ ⌉ is the round-up function; and all of the U's denote PRNs.) - CORRECT ANSWER>>>>⌈ 6 U 1 ⌉ + ⌈ 6 U 2 ⌉ If U is Unif(0,1), how can we simulate a Geom(0.6) random variate? - CORRECT ANSWER>>>>⌈ ln ( U ) / ln ( 0.4 ) ⌉ ⌈ ln ( 1 − U ) / ln ( 0.4 ) ⌉ Consider four observations from some unknown distribution, X 1 = 1.5, X 2 = − 3.7, X 3 = 2.7, and X 4 = 0.6. What is the fourth order statistic, which we denoted by X ( 4 ) in class? - CORRECT ANSWER>>>>2. merely means the largest of the sample of 4 observations.

Suppose that U and V are PRNs. Let X = U + V. Simulate this 5000 times, and draw a histogram of the 5000 numbers. What p.d.f. does the histogram look like? - CORRECT ANSWER>>>>Triangular By the lesson notes, we know that the 5000 Xi's are all Triangular(0,1,2). Since we have a histogram of 5000 of these, it really ought to look like a triangular p.d.f. Thus, the answer is (c). Suppose that U 1 , U 2 , ... , U 24 are i.i.d. PRNs. What is the approximate distribution of X = 5 + 3 ∑ i = 1 24 U i? - CORRECT ANSWER>>>>Nor(41,18) In general, the majorizing function t ( x ) is itself a p.d.f. f ( x ). - CORRECT ANSWER>>>>False the majorizing function generally integrates to a number greater than 1, and so it cannot be a legitimate p.d.f. Suppose that X is a continuous RV with p.d.f. f ( x ) = 30 x 4 ( 1 − x ), for 0 < x < 1. What's a good method that you can use to generate a realization of X? - CORRECT ANSWER>>>>Acceptance- Rejection Consider the constant c = ∫ R t ( x ) d x = 5. On average, how many iterations (trials) will the A-R algorithm require? - CORRECT ANSWER>>>> Powered by TC PDF ( www.tc pdf.org)