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Unformatted text preview: BME 335 Exam I Spring 2008 exam# BME 335
Exam I
Spring 2008 I understand that the exam proctors will watch for violations of the exam rules and that
offending students will be reported to the Dean of Students. I understand that Scholastic
Dishonesty can result in an automatic F in the course. I understand the exam rules. I know that I have 75 minutes to complete this exam 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 with the exam by the instructor. , No electronic devices such as calculators or cell phones may be used during the exam time. I should not write my name or any other identifying information on any portion of the
exam other than this face page. Writing my name or any other identifying information on
any portion of the exam other than this face page will be interpreted as an attempt to
gain any unfair advantage and may result in automatic failure of the exam. My answers must be leginy written and clearly demonstrate my thought process in order
to earn full credit on the exam. I know that mathematical notation has been consistently
used in this course and that I must follow that same 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 10
simplify fractions or combinatorics, e.g., 1:126 and [3) are acceptable answer forms.
+
Print Name
Sign Name Date BME 335 Exam I Problem 1: / 2
Problem 2: / 2
Problem 3: / 2
Problem 4: w / 2
Problem 5: / 4
Problem 6: / 4
Problem 7: / 6
Problem 80: __ /1
Problem 8b: / 1
Problem 8c: / 2
Problem 8d: / 2
Problem 8e: / 4
Problem 8f: / 4
Problem 9: / 5
Problem 10: / 5
Problem 110: m/ 1
Problem 11b: / 2
Problem 11c: / 1
Problem 11d: / 2
Problem 11e: / 4
Problem 111‘: / 5
Problem 119: / 2
Problem 12o: / 6
Problem 12b: / 10
Problem 12c: / 6
Problem 13: / 10
Problem 14: / 5 EXAM SCORE: / 100 Spring 2008 exom# BME 335 Exam I Spring 2008 exom# Counﬁng Formuios Common PMF/PDFs PMF/PDF — — 2
1 ifk=a,a+1,...,b “5 W
PX(k)= b—a+1 2 12
 0 otherwise
{ ' p 1701)) _ _ if“ A
p"(k)=eAE k=0’1"”  A
b u 2
1 if a s x s b a + (b ‘0
fx (x) b — a 2 12
0 otherwise
f M“ if x 2 0 1 i
2
0 otherwise A A
4mm 2 Ax):
13(35): mﬁwwm 7,. a .,, , mm“ 7” a , mémﬁwt” .. . .774 Set; 3.3 Normal Random Variables 155 The standard normal table. The entries in this table provide the numerical values
of @(y) = P(Y S y), Where Y is a standard normal random variable, for y between 0
and 3.49, For example, to ﬁnd @[1.7l), we look at the row corresponding to 1.7 and
m the column corresponding to 0.01, so that @(111) 2 .9564. When y is negative, the value 0f<l1(y) can be found using the formula @(y) :2 1  @(wy). BME 335 Exam I Spring 2008 exam# (1) Define random variable (2 poinTs) (2) Random variables X and Y are independenT if for all x and y... (2 poinTs) (3) Define The marginal PMF of The random variable X from The joinT discreTe PMF of X and Y.
Use maThemaTical noTaTion, noT narraTive TexT. (2 painTs)  (4) Define The condiTional PMF of The random variable X condiTioned on The random variable Y,
in Terms of The joinT PMF. Use maThemaTicol noTaTion, noT narraTive TexT. (2 poinTs) (5) STaTe DeMorgan's Laws in seT noTaTion (4 poinTs) BME 335 Exam I Spring 2008 exam# ' (6) State the total probability relationship in terms of diagnostic testing: D, diseaSe: +, positive
test; and —, negative test. (4 points) (7) Suppose that a lesion can be benign (B), mild dySplasia (M), or severe dysplasia (5). State
Bayes' theorem for calculating the probability of severe dysplasia given the lesion radius (R).
Your answer has to in terms of priors and likelihoods, i.e., the denominator can't simply be P02).
(6 point) BME 335 Exam I Spring 2008 exam# ' (8) In The TradiTion of gameshows like LeT's Make a Deal and Deal or No Deal, Prof. Markey decides To SeT up a gameshow aT The BME fall welcome gaThering. In Win Mia‘s Money, There are
Three envelopes: red, blue, and green. One envelope has $1, one has $10, and one has $100. (So) If n = 3 and k = 3, how many ways could The differenT denominaTions be puT inTo The
envelopes? (circle one correcT answer: 1 poinT) (i) nk
(H) [n +: 4) n!
(m) (n 46)! (2:) (8b) Finish The Table lisTing The ways in which The differenT denominaTions could be puT inTo The
enveloes. (1 oinT) (8c) Supp05e you selecT The red envelope. WhaT is The expecTed value of The conTenTs of The
red envelope? SeT up only, buT all sTeps musT be shown To earn full crediT. (2 poinTs) (8d) Suppose you selecT The red envelope. WhaT is The probabiliTy ThaT The red envelope
conTains $100? SeT up only, buT oll sTeps musT be shown To earn full crediT. (2 poinTs) BME 335 Exam I Spring 2008 exam# (8e) Suppose ThaT you selecT The red envelope and ThaT Prof. Markey randomly SelecTs one of
The oTher envelopes To open. She opens The blue envelope and reveals ThaT iT does noT conTain
The $100. CompleTe The calculoTions below To find The probability ThoT The red (your envelope)
vs. green (The oTher envelope) conTains The $100, given The new informaTion abouT The conTenTs
of The blue envelope. (4 poinTs) P(R = $100B :2 $100) = ﬂT—ﬂﬂ—STL) = :—?—~ = —/_ P(G = $100B ¢ $100) = ££——ﬂ_—““m_m_) _ :4 a ———/w P<___> / (8f) Suppose ThaT you SelecT The red envelope and ThaT Prof. Markey selecTs one of The oTher
envelopes To open. BuT, raTher Than randomly selecTing which envelope To open, Prof. Markey
deliberarely opens The remaining envelope Tho?“ she knows does noT conmr'n The $100. SuppOSe
ThaT she opens The blue envelope and reveals ThaT iT does noT conTain The $100. CompleTe The
calculaTion below To find The probabiliTy ThoT The red (your envelope) vs. green (The oTher
envelope) conTains The $100, given The new informoTion abouT The conTenTs of The blue envelope.
(4 poinTs) P_(R z $10010penblue) = P(0penblueR = $100)P(R = $100)
P(0penbluelR = $100)P(R = $100) + P(apenblueB = $100)P( )+ P(0penbluel )P(G = $100) P(G = $1000penblue) = P(——)P( BME 335 Exam I Spring 2008 exam# (9) There are 21 students in the Monday lab session. Assuming no one has a leapyear birthday,
i.e., no birthdays on Feb 29th. and that all days are equally likely for birth, Set up the calculation
for finding the probability that each person has a distinct birthday. Hints: What is the size of
the sample space? How many possible birthdays for the first person? What about the second
person if different from the first? (5 points) Lab Num. Probability that Session Students each has
distinct
birthday m
F11
M2
T2 (10) How many ways different "words" can be formed by rearranging the letters in “engineering”? (In this context, “words” are just different strings of letters, they don‘t have to
be real words.) (5 points) BME 335 Exam I Spring 2008 exam# (1 1) Suppose that in the SWE Mr. Engineering pageant, there are 15 contestants. Suppose that
in the swimwear competition, one of faculty judges assigns scores in two steps. First, she
decides to give a nonzero score with probability 30% and zero points with probability 70%. For
any contestant she decides to give a nonzero score, she gives him 10 points. (1 In) What is the name of the common random variable that we use to model the number of
contestants that get a score of zero? (circle one correct answer: 1 point) (i) Bernoulli (ii) binomial (iii) Poisson (iv) geometric (11b) What is the average number of contestants that the judge will decide to give a score of
zero? Set up only, but all steps must be shown to earn full credit. (2 points) (11c) What is the average number of points that the judge assigns to a single contestant? Set
up only, but all steps must be shown to earn full credit. (1 points) (11d) What is the average number of points that the judge assigns in the entire swimwear
competition? Set up only, but all steps must be shown to earn full credit. (2 points) (11a) What is the probability that the judge assigns a final score of 10 points to less than 2 of
the contestants? Set up only, but all steps must be shown to earn full credit. (4 points) 10 BME 335 Exam I Spring 2008 exam# (11f) SkeTch a diagram of The PMF of The random variable ThaT could be usad To model The
number of conTesTanTs raTed unTil The judge firsT assigns a score of 10. Label The diagram clearly. (5 poinTs) (1 lg) Suppose ThaT The judge's raTing of a given conTesTanT was affecTed by how she raTed The
previous conTesTanT. Would The above calculaTions be The same? Why or why MT? (2 poinTs) 11 BME 335 Exam I Spring 2008 exam# (12) CompleTe The calculaTions below To use The linear Transformation properTy of normal random variables and The CDF of The sTandard normal To deTermine whaT percenT of values fall
wiThin TWO sTandard deviaTions of The mean. (12a) (6 points) G ~ N(15,3) = L—_ H N H
P(15—2*3<G<15+2*3)=P(9<G<21)
P(9<G<21)2P(:—:~—:<H<::—:)
P(9<G<21)=P(H<w_)—P(H<_m)
P(9 < G <21) = c1>(_)—c1)(_) P(9 < G <21) =(D(_)—(1—<I)(_)) P(9 < G< 21) = ___—(1—____)
P(9<G<21)=0.9544 So, 95% of the values are Within i 2 standard deviations of the mean. 12 BME 335 Exam I Spring 2008 exam# (12b) Now, Try iT again wiTh anoTher normal RV wiTh a differenT mean and variance. Use The
linear TransformaTion properTy of normal random variables and The CDF of The sTandard normal
To deTermine whaT percenT of values fall wiThin Two sTandard deviaTions of The mean. (10 poinTs) Q ~ N(2o,5) 13 BME 335 Exam I Spring 2008 exam# (12c) Draw a diagram To explain The caicula‘rions performed in 120&b. Label The diagram dearly.
(6 points) 14 BME 335 Exam I Spring 2008 exam# (13) Fill in The missing informaTion To confirm The formulas for The mean and variance of a
1 conTinuous uniform random variable wiTh PDF of fX(x) = b_a a 5 x S b (10 poinTs)
0, otherwise E[X]=:£x dx E[X]=T_biadx (in Terms of X) (in Terms of a and b) 15 exam# BME 335 Exam I Spring 2008 (14) Draw a diagram of The CDF corr65ponding To The PDF of fx(x) ={ diagram clearly. (5 poin’rs) 1 0, b a asxs‘biabel’rl'ua otherwise 16 ...
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