1. Here is a distribution. y p(y) A.(5) Draw a graph of this distribution. Solution:

Transcription

1 ISQS 5347 Final Exam. Instructions: Open book. No loose leaf notes. No electronic devices. Put all answers on the paper provided to ou. Points (out of 200) are in parentheses. 1. Here is a distribution. p() A.(5) Draw a graph of this distribution. p_ B.(5) Calculate E(Y) for this distribution. E(Y) = = = C.(10) Explain the meaning of the ou calculated in 1.B. using the Law of Large Numbers. Refer ver specificall to (i) the distribution shown above, and (ii) the number calculated in 1.B. in our answer. If data values Y 1,, Y n are produced as iid from the distribution shown above, and the average is Y = (Y 1 + Y Y n )/n is calculated, then Y will be approximatel equal to 1.9. For larger n, the approximation is better.

2 2. Suppose Y ~ U(0, θ ), the uniform distribution with unknown upper limit θ. 2.A.(5) Draw a graph of this distribution. Annotate both axes. p() 1/θ 0 θ/2 θ 2.B.(5) Explain wh E(Y) = θ/2 using the graph of 2.A. and the point of balance concept. See the graph. The densit function, if it were a cardboard cut-out, would balance at the value θ/2. If the point were higher than θ/2 it would fall to the left; if it were lower than θ/2 it would fall to the right. 2.C.(10) Using the fact that E(Y) = θ/2, demonstrate that θˆ= 2Y is an unbiased estimator of θ. Use (i) the definition of unbiasedness and (ii) linearit and/or additivit properties of expectation for each logical step of our demonstration. B definition of unbiasedness, θˆ is an unbiased estimator of θ if E(θˆ) = θ. So let s calculate E(θˆ): E(θˆ) = E(2Y) (b substitution) = 2E(Y) (b the linearit propert of expectation) = 2(θ/2) (using the fact that E(Y) = θ/2) = θ (b algebra). Thus, E(θˆ) = θ, which implies that θˆ is an unbiased estimator of θ.

3 3. Twent-five students will be asked, what is our favorite beverage? The will be asked to choose between Soda, Coffee, and Tea. A hpothetical realization of the data could look like 1 = Coffee, 2 = Coffee, 3 = Soda, 4 = Tea,, 25 = Soda, but man other realizations are also possible. 3.A.(10) Give a model for how the data Y 1, Y 2,, Y 25 will appear, in the absence of an other information about the students. Be as specific as ou can. Solution. A good model is Y 1, Y 2,, Y 25 ~ iid p(). Here, p() has the following form: p() Soda π 1 Coffee π 2 Tea π 3 Total: 1.00

4 3.B.(15) Suppose data on student US citizenship, X, either Yes or No, is available. A hpothetical realization of the data in this case could look like (x 1, 1 ) = (No, Coffee), (x 2, 2 ) = (Yes, Coffee), (x 3, 3 ) = (No, Soda), (x 4, 4 ) = (No, Tea),, (x 25, 25 ) = (Yes, Soda), but man other realizations are also possible. Give a regression model for how the data Y 1, Y 2,, Y 25 will appear, given the data X 1 =x 1, X 2 =x 2, X 25 =x 25. Be as specific as ou can. Solution. A good model is Y 1 X 1 = x 1, Y 2 X 2 = x 2,, Y 25 X 25 = x 25 ~ ind p( x). Here, p( x) has the following form: Soda Coffee p( X =Yes) π 1,es π 2,es π 3,es Tea Total: 1.00 Soda Coffee Tea p( X =No) π 1,no π 2,no π 3,no Total: (15) Nature favors continuit over discontinuit describes conditional distributions that produce data. Use the example Y = average dail food expense and X = annual income. Draw and annotate graphs of conditional distributions p( x) that clearl show what nature favors continuit over discontinuit means in this example.

5 densit Food Expense In the graph, the solid and dashed lines are distributions of food expense for people whose incomes differ ver little, eg, the solid line might be the food expense distribution for people with Income=30,000 and the dashed line might be the food expense distribution for people with Income = 31,000. These distributions differ ver little, demonstrating the idea that Nature favors continuit over discontinuit. The dotted line, on the other hand, shows a hpothetical food expense distribution for people with Income = 100,000. So the distributions differ greatl when the X differs greatl, but the distributions are similar when the X values are similar. 5.(25) Suppose Y 1, Y 2,, Y n ~ iid p(), where p() is a non-normal distribution with finite variance. Let S denote the sum of the Y s, S = Y 1 +Y 2 + +Y n. The Central Limit Theorem states that S has an approximatel normal distribution. Describe a simulation stud that demonstrates that the sum, S, of n=100 values produced as iid from the Poisson distribution with λ = 1, has an approximate normal distribution. Structure our answer as follows:

6 Step 1:. Step 2:. Use the concept of a q-q plot in our answer. Do not write R code. Instead, just sa what ou are doing at each step. Be ver clear in stating what result of our simulation stud and q-q plot will show that S has approximatel a normal distribution. Step 1: Generate 100 random numbers from the Poisson distribution with λ = 1, Y 1,,Y 100. Step 2: Calculate the sum, S= Y 1 + Y Y 100. Step 3: Repeat steps 1 and 2 100,000 times, getting values S 1, S 2,,S ,each a sum of 100 numbers. Step 4: Plot the sums S 1, S 2,,S using a q-q plot. If the q-q plot is approximatel a straight line, then this will demonstrate that the distribution of S is approximatel a normal distribution. 6.(25) If data Y 1, Y 2,, Y 100 are produced as iid from the N(0,1) distribution, then Y = (Y 1 + Y Y 100 )/100 is also a random variable. You know from probabilit theor that the standard deviation of Y is σ/ 100 = 1/10 = 0.1. Describe, step b step, a simulation stud that shows that the standard deviation of the average Y = (Y 1 + Y Y 100 )/100 is approximatel 0.1. There must be no calculation of 1/10 = 0.1 in our simulation stud. Instead, state what in the simulation will give the value of approximatel 0.1, demonstrating that the standard deviation of Y is 0.1. Structure our answer as follows: Step 1:. Step 2:. (more steps until done) Do not write R code. Instead, just sa what ou are doing at each step. Be ver clear in stating what result of our simulation stud will show that the standard deviation of Y = (Y 1 + Y Y 100 )/100 is approximatel 0.1. Step 1: Generate 100 random numbers from the N(0,1), Y 1,,Y 100. Step 2: Calculate the average, Y = (Y 1 + Y Y 100 )/100.

7 Step 3: Repeat steps 1 and 2100,000 times, getting values Y 1, Y 2,, Y , each an average of 100 numbers. Step 4: Calculate the standard deviation of the 100,000 values in Step 3. This number will be approximatel (15) A test of a restricted data-producing model gives a likelihood ratio chi-square statistic of 21.2 and a corresponding p-value p = The probabilit 0.02 can be understood loosel as 2 out of 100. Explain what 2 out of 100 means here; specificall, what does 2 refer to, and what does 100 refer to? In our answer include all of the following terms: (i) restricted data-producing model, (ii) repeated samples, (iii) random likelihood ratio chi square statistic, (4) If the data are produced b the restricted data-producing model, then each sample will ield a random likelihood ratio chi square statistic. In 100 such repeated samples, approximatel 2 out of 100 will produce a likelihood ratio chi square statistic that is greater then Man statistical procedures assume normalit of the distributions p() that are assumed to produce the data. But real data that we can analze are never produced b normal distributions. Consider our example Y = a person s claimed average dail food expense. 8.A.(5) Explain wh the p() that produces these Y data values must be a discrete distribution. The data we analzed were rounded to the nearest dollar. Hence the cannot fill a continuum. But no matter how the are collected no data we humans can analze can possibl fill a continuum due to finiteness of our measuring instruments. 8.B.(5) Explain wh the p() that produces these Y data values must be a skewed (non-smmetric) distribution. The distribution is bounded below b 0 but unbounded at the high end. Thus there is a possibilit to observe an occasional extremel large value, but no possibilit to observe an occasional extremel small value. 9.(5) An R command is x = seq(0, 10,.01) What is x? The list of values 0.00, 0.01, 0.02,, 9.99,

8 Multiple choice R questions. (Four points each) 10. What does the R function rnorm(100) do? A. Generates 100 normall distributed data values. B. Computes the densit of the normal distribution at = 100. C. Finds the q-q plot from the normal distribution with 100 data values. D. Graphs the normal densit function from 0 to Which R command allows ou to read data into the program? A. read.data B. data.read C. all.in D. read.csv 12. Which R command draws a scatterplot of data pairs (x, )? A. scatter(x,) B. plot(x,) C. pairs(x,) D. all(x,) 13. Wh are data generated from R s computer random number generators independent? A. Because that is the wa computer random number generators work. B. Because the are from the same distribution. C. Because the are identicall distributed. D. Because the data depend on R s random number generator seed value. 14. Which command generates a sample from the bootstrap distribution of the data in the R list? A. bootstrap(,length(), replace=t) B. bootstrap(, length(), replace=f) C. sample(, length()), replace =T) D. sample(, length()), replace =F) 15. Which R function does Baesian analsis? A. RBaes B. BaesR C. MCMCRegress D. MaxLik 16. Wh is == used in R? A. to allow roundoff errors B. to compare different quantities C. to force items to be equal even when the are not D. to generate random numbers

9 17. What does tpe=h do in the plot function? A. increases the height of the plot B. makes the plot a needle plot C. makes the plot character a square rather than a circle D. connects the points with straight lines 18. What is the purpose of jitter in R, as in plot(jitter(x), jitter())? A. To make the distribution look more normal B. To make the relationship closer to linear C. To keep points from being plotted on top of each other D. To solve the log likelihood function for the maximum likelihood estimates 19. Data frames, lists, fitted models, and matrices are examples of in R. A. objects B. variables C. observations D. functions