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Unformatted text preview: BME 335 Exam I Fall 2006 BME 335
Exam I
Fall 2006 I understand that the exam proctors will watch for violations of the exam rules and that
offending students will be reported to theDean of Students. I understand that Scholastic
Dishonesty can result in an automatic F in the course. I understand the exam rules. I know that O I have 75 minutes to complete this exam: start 5 PM & end 6:15 PM I may not communicate in any way with anyone other than exam proctors during the exam
time I may not consult textbooks, notes, or other reference material during the exam time
other than the reference material provided on the next two pages. No electronic devices such as calculators or cell phones may be used during the exam
time. I know that mathematical notation has been consistently used in'this course and that I
must follow that some convention on the exam. I understand that numerically “right"
answers that don't show my problem solving process will not earn full credit. For example,
in a “counting problem", I need to specify in my answer if order matters, if there is sampling with replacement, etc. I do not have to simplify fractions or combinatorics, e._q.,
5 + 2 12+16 the steps up to the point that either I would punch numbers into my calculator or employ
basic calculus to arrive at a numerical answer. I should not write my name or other identifying information on any portion of the exam
other than the blanks indicated below. 10
and (3) are acceptable answer forms. Unless otherwise indicated, I must show all Print Name
Sign Name Date BME 335 Exam I Fall 2006 Counﬁng Formulas Variance (b—a)(b——a+2) 12
10) otherwise
if x 2 0 otherwise BME 335 Exam I Fall 2006 (1) Define random variable (2 poin’rs) (2) Define probabili‘ry mass func’rion (2 poin’rs) (3) Define probability densiTy function (2 poinfs) (4) Define cumulative disfribufion func’rion (2 poim‘s) (5) Define conditional probabilify (2 poin’rs) (6) Define independence (2 points) BME 335 Exam I Fall 2006 (7) Consider a molecule ThaT has 10 binding siTes of Type A and 20 binding siTes of Type B. If 5
ligands are bound, whaT is The probabiliTy ThaT exachy 2 are bound aT Type A siTes? (3 poinTs) (8) IndicaTe which random variable could besT be used To model The following siTuaTions. You
musT expiain your choice in order To earn crediT. (a) The number of cases of Ebola reporTed each year in a given counTry. (1 poinT) (b) The number of cases of influenza in a given preschool classroom in December. (1 poinT) (c) The Time unTil an xray machine in a hospiTal breaks down. (1 poinT) (9) Considering counTing problems wiTh some fixed n and k. (a) Circle The siTuaTion ThaT would yield more possibiliTies: (1 poinT)
ordered unordered (b) Circle The siTuaTion ThaT would yield more possibiliTies: (1 poinT)
replacemenT no replacemenT (c) Provide 1«2 senTences To jusTify your answers To a 6: b. (3 poinTs) BME 335 Exam I Fall 2006 (10) For each of The following Venn diagrams, use 581' noTaTion ‘ro describe The shaded region. (6
poim‘s) BME 335 Exam I Fall 2006 (11) For each PMF/PDF shown, indico’re The name of The common RV depicted and approximate
The parameTers. Provide a few sentences justifying your answer. (24 poin’rs) 0.5  0.4 Nome:
WHO 3:: Poronwefer‘ Volue(s):
0.1 JUSTIfICGTIOr}?
o Nome:
Parome’rer Volue(s):
Jus’rifico’rion: 0123456789 X Name:
Parameter Vofue(s):
JusTifica’rion: §A_, ,,
3456789
k BME 335 Exam I Fall 2006 (11 continued) Name:
Parame'rer Vaiue(s):
Jus’rificafion: 0.35
0.3
0.25
Mk) 0%
0.1
0.05 W" '1
i
i 0123456789
k Name:
Parameter Value(s):
Justification: 0123456789
x 0.25 1 Name:
0.2 Parameter Value(s)1
0'15 JusTifica’rion:
x k
p ( ) 0.1
0.05 0123456789
k BME 335 ‘ Exam I Fall 2006 (12) A radiologist examines lesions on a MRI exam and indicates the likelihood of malignancy on
a continuous scale of 1 to 10, where 1 indicates that the abnormality is very unlikely to be
cancer and 10 indicates that the abnormality is very likely to be cancer. Based on a large
experiment, it is determined that for lesions which are actually cancer, the radiologist's
estimate of the likelihood of disease follows a normal distribution with mean of 7 and standard
deviation of 1. Similarly, for benign lesions, the radiologist’s estimate of the likelihood of
disease follows a normal distribution with mean of 3 and variance of 4. Suppose that a threshold
is applied to the radiologist's estimate of the likelihood of disease such that if the radiologist
rates the lesion above 4, then it will be considered to be positive and further workup will be
done. Likewise, if the radiologist rates the lesion below 4, then it will be considered negative
and no further workup will be done. (a) What is the sensitivity? Work this all the way to a number. Clearly show all steps of
your calculation. You will probably find it helpful to sketch the PDF. (10 points) BME 335 Exam I Fall 2006 (12 conﬁrmed)
(b) WhaT is The specificity? Work This all The way To a number. Clearly show all sTeps of your calculaTion. You will probably find iT helpful To skeTch The PDF. (10 poinTs) BME 335 Exam I Fall 2006 (12 conTinued)
(c) Suppose ThaT The radiologisT were To aSSess a populaTion of paTienTs in which The
prevalence of disease is 3%. WhaT is The posiTive predicTive value? (SeT up only) (8 poinTs) (d) Suppose The radiologisT were To assess a second populaTion in which The prevalence of
disease is 30%. Would The posiTive predicTive value be lower, abouT The same, or higher Than whaT you seT up in (c)? Explain your answer. (1 poinT) 10 BME 335 Exam I Fall 2006 (13) You are conducting a series of experiments to characterize the efficacy of some
biopolymers for drug delivery. In your studies, you vary two properties of the biopolymers,
which we will refer to simply as A and B, to a few discrete levels of each. The joint PMF
below deicts the distribution of the effective bio0 mers over the nuroerties studied. —_—
4/20 3/20 2/20 1/20 3/20 2/20 2/20
1/20 1/20 1/20 0/20 (a) What is the PMF for the different levels of property A, regardless of the level of
property B? (Le, the marginal PMF for A) (2 points) (b) What is the conditional PMF for property B, given that property A is fixed at level3? (2
points) (c) Based on these data, are the properties A and B independent? Explain your answer. (2
points) 11 Fall 2006 BME 335 ExomI
1/2 x21
5/6 y=1
1/3 x=2
(14) Suppose ‘rhoTpY(y)= 1/6 y=2 ("Id leY(xiy=l)= N6 x_3 and
0 otherwise — .
O otherwzse
1/2 x=3
U3 x=4 pxl"(x[y&2)ﬁ 1/6 x=5 0 otherwise (a) Wha’r is E[Y]? (2 points) (b) What is FY01)? (2 points) (c) Suppose ’rhoT Z = Y2. Wha’r is pz(z)? (2 points) (d) Suppose fho‘r W = 3Y+2. Who’r isE[W]? (2 points) (e) What is E[XY =1]? (2 poin’rs) (f) Who’r is AU)? (2 points) 12 ...
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