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Stat 330X Exam II October 22, 1999 Prof. Vardeman
- Some jobs submitted for processing on a particular CPU have fatal programming errors, while others do not. Suppose that the long run fraction of jobs with fatal programming errors is : ú fi!&.
a) Find (under appropriate model assumptions) the probability that among the next 10 jobs submitted there are less than 3 with fatal errors.
b) One begins monitoring the processing of jobs and lets
\ ú the number of jobs processed before the first with a fatal error.
Find (again under appropriate model assumptions) T “\ ü #&”fi
c) One begins monitoring the processing of jobs and lets
] ú the total number of jobs without errors processed before the #nd job with a fatal error.
Find (under appropriate model assumptions) E ] and Var].
- Hits on a popular Web page occur according to a Poisson Process with rate - ú "!hits/min. One begins observation at exactly noon tomorrow (WOI standard time).
a) Evaluate the probability of #or less hits in the first minute.
b) Evaluate the probability that the time till the first hit exceeds 10 seconds.
c) Evaluate the mean and the variance of the time till the 4th hit.
d) Evaluate the probability that the time till the 4th hit exceeds 24 seconds.
e) The number of hits in the first hour is Poisson with mean 600. You would like to know the probability of more than 650 hits. Exact calculation isn't really feasible. So approximate this probability and justify your approximation.
- Suppose that Y fl Y" # and Y (^) $are independent Uniform –!fl "—random variables and distributional properties of ] ú min–Y fl Y fl Y —fl [ ú (^) " # $ max–Y fl Y fl Y —" # $ and V ú [ Å ]are of interest. Attached to this exam is a printout from a Minitab session conducted to study these variables. (On that printout "RMinimum" and "RMaximum" are minimum and maximum values in a given row.)
a) Use the printout to deduce approximate values for the following:
E [ ∏ __________ E ] ∏ __________ EV ∏__________
Var [ ∏ __________ Var ] ∏ __________ VarV ∏__________
T “V û fi&” ∏ __________
b) Do your approximate values of Var [ and Var]sum to something close to your approximate value of VarV? Do you expect them to? Explain.
- A continuous distribution of interest has distribution function
J –C— ú
! C ü! " Å –" Å C—! ù C ù " " C † "
if if if
$
Describe as completely as possible how you would simulate a realization of ] with this distribution.
Minitab Printout for Stat 330X Exam II, Problem 5
MTB > Random 1000 C1-C3; SUBC> Uniform 0.0 1.0. MTB > RMinimum C1 C2 C3 C4. MTB > RMaximum C1 C2 C3 C5. MTB > Let C6 = C5 - C MTB > Describe C4 C5 C6.
Descriptive Statistics
Variable N Mean Median TrMean StDev SE Mean C4 1000 0.25134 0.21293 0.23860 0.19324 0. C5 1000 0.75152 0.79377 0.76452 0.19303 0. C6 1000 0.50017 0.50961 0.50039 0.21395 0.
Variable Minimum Maximum Q1 Q C4 0.00002 0.90188 0.09468 0. C5 0.12254 0.99850 0.63536 0. C6 0.01594 0.98028 0.33519 0.
MTB > GStd.
- NOTE * Character graphs are obsolete.
MTB > Histogram C6; SUBC> Start .025; SUBC> Increment .05.
Histogram
Histogram of C6 N = 1000 Each * represents 2 observation(s)
Midpoint Count 0.0250 4 ** 0.0750 22 *********** 0.1250 22 *********** 0.1750 42 ********************* 0.2250 59 ****************************** 0.2750 53 *************************** 0.3250 73 ************************************* 0.3750 71 ************************************ 0.4250 69 *********************************** 0.4750 75 ************************************** 0.5250 70 *********************************** 0.5750 93 *********************************************** 0.6250 77 *************************************** 0.6750 71 ************************************ 0.7250 68 ********************************** 0.7750 43 ********************** 0.8250 39 ******************** 0.8750 30 *************** 0.9250 15 ******** 0.9750 4 **
