ECEn 370 Final
Name:
RED- You can write on this exam.
ECEn 370
Introduction to Probability
Section 001
Final
Winter, 2011
Instructor
Professor Brian Mazzeo
Closed Book - You can bring TWO 8.5 X 11 sheets of handwritten notes (both sides)
Graphing or Scientic Calculator Allowed
Three-Hour Time Limit
IMPORTANT!
•
WRITE YOUR NAME on every page of the exam.
•
All questions are free response. SHOW ALL WORK.
•
Do not discuss the exam with other students.
•
I agree to abide by the Honor Code.
•
Signature: ____________________________, Date: ___________________________
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ECEn 370 Final
Name:
1. (5 points) A batch of fty items is to be inspected by testing k randomly selected items. If one of the k items is
defective, the batch is rejected. What is the minimum number of items, k, you should test to make sure that there
is at least a probability of 0.90 of detecting a batch containing exactly 10 defective items?
2. (5 points) Three hundred students, including Joe, Jane, and Julia, are to be split into three classes of equal size,
and this is to be done at random. What is the probability that Joe, Jane, and Julia end up in the same class?
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ECEn 370 Final
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3. (5 points) You place 110 bricks end-to-end to make a barrier around a garden. Each brick length is described
by a uniform random variable with length ranging from 9.5 to 10.5 inches. What is the probability that the total
length of the barrier is between 1093 and 1105 inches. Use the Standard Normal Table provided.
4. (5 points) Suppose that X , Y , and Z are independent random variables. E[X] = 4, E[Y ] = 2, and E[Z] = 3.
var(X) = 3, var(Y ) = 2, and var(Z) = 1. Compute the following expressions.
a) E[2X + X 2 ] =
b) E[3XY Z − Y ] =
c) var(−2X + 3Z) =
d) var(Y − X) =
e) E[XY 2 ] =
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ECEn 370 Final
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5. (6 points) You have two ten-sided dice numbered from 1 to 10. Each face has an equal probabilitiy of appearing
during a roll (uniformly distributed). The outcomes, representing each die, are the random variables X and Y .
a) (2 pts) Find P (max(X, Y ) = 4).
b) (2 pts) Find P (min(X, Y ) = 6).
c) (2 pts) Find P (X + Y = 10 | X ≤ 2) .
6. (5 points) A binary signal S is transmitted, and we are given that P (S = 1) = 0.3 and P (S = 0) = 0.7. The
received signal is Y = N + S , where N is normal noise, with zero mean and unit variance, independent of S . What
is the probability that S = 1, as a function of the observed value 0.7 of Y ?
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ECEn 370 Final
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7. (10 points) A stock market trader buys 10 shares of stock A and 20 shares of stock B. Let X and Y be the price
changes of A and B, respectively, over a certain time period, and assume the joint PMF of X and Y is uniform over
the set of integers satisfying
−2 ≤ x ≤ 1,
−1 ≤ y − x ≤ 1
a) (3 pts) Find the mean of the trader's prots (or losses if negative).
b) (3 pts) Find the covariance of X and Y .
c) (3 pts) Find the correlation coecient of X and Y .
d) (1 pt) Are the variables X and Y uncorrelated, positively correlated, or negatively correlated?
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ECEn 370 Final
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8. (27 points) (Based on HW 6.14) Beginning at time t = 0, you start eating candy bars, one at a time. You start
eating the next candy bar immediately upon nishing each candy bar. Each new candy bar is selected independently
by an equally likely choice between a MarkovWay and a BernoulliRuth. The time it takes to eat a candy bar, X ,
of any particular candy bar is a random variable with the following PDFs:
(
for MarkovWay candy bars: fX (x) =
2e−2x , if x ≥ 0,
0,
otherwise;
(
for BernoulliRuth candy bars: fX (x) =
3e−3x , if x ≥ 0,
0,
otherwise;
a) (3 pts) Find the expected time until the rst candy bar is eaten.
b) (3 pts) Find the probability that there are no candy bars eaten before time t.
c) (3 pts) Given that there are no candy bars eaten until time t, determine the conditional probability that the rst
bar eaten is a MarkovWay candy bar.
d) (3 pts) Find the variance of the time until the rst candy bar is eaten.
e) (3 pts) Find the probability that the 14th candy bar eaten is also the 4th MarkovWay candy bar eaten.
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ECEn 370 Final
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f) (3 pts) Up to and including the 12th candy bar eaten, what is the probability that a total of exactly 5 MarkovWay
candy bars have been eaten?
g) (3 pts) Determine the transform associated with the time until the 11th candy bar is eaten.
h) (3 pts) Determine the probability that the total eating time for eating the rst two BernoulliRuth candy bars is
longer than the time taken to eat the rst MarkovWay candy bar. You can leave it in integral form.
i) (3 pts) Suppose that the process terminates as soon as a total of exactly 14 candy bars have been eaten. Determine
the expected value and variance of the total time it took to eat BernoulliRuth candy bars while the process is in
operation.
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ECEn 370 Final
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9. (6 points) Suppose I have the following transforms for two independent random variables, X and Y :
MX (s) =
1 7s 1 3s 1 6s
e + e + e ,
2
6
3
MY (s) =
1 2s 3 3s
e + e
4
4
(2 pts) What is E[X 2 ]?
(2 pts) What is pX (3)?
(2 pts) What is E[X + Y ]?
10. (8 points) Let X and Y be independent random variables that are uniformly distributed on the interval [0, 1].
What is the PDF of the random variable Z = Y /X .
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ECEn 370 Final
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11. (10 pts) (Based on HW 5.5) Let X1 , X2 , . . . be independent random variables that are uniformly distributed
over [-1, 1]. Show that the sequence Y1 , Y2 , . . . converges in probability to some limit, and identify the limit, for
each of the following cases:
a) (3 pts) Yn = Xn /n
b) (3 pts) Yn = max {X1 , . . . , Xn }
c) (4 pts) For the following random variable, Yn , it has a distribution given by

1

1 − n , for y = 0,
P (Yn = y) = n1 ,
for y = n2 ,


0,
otherwise
Find E [Yn ]. Does Yn converge in probability?
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ECEn 370 Final
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12. (12 points) Consider the following Markov chain with the indicated single-step transition probabilities.
1
0.5
0.3
1
0.3
0.2
2
0.4
3
0.2
0.3
0.7
4
1
5
0.1
a) (4 pts) Given that you are in the 4,5 recurrent class, what is the steady-state probability of being in state 5?
b) (4 pts) Given that you are in state 2, what is the probability of being absorbed in state 1?
c) (4 pts) What is the expected number of transitions until absorption for states 2 and for state 3?
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ECEn 370 Final
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13. (14 points) (Similar to Problem 3.23 from HW) Let the random variables X and Y have a joint PDF which is
uniform over the triangle with vertices at (0,0), (0,4), and (1,1).
a) (2 pts) Find the joint PDF of X and Y .
b) (3 pts) Find the marginal PDF of X .
c) (3 pts) Find the conditional PDF of Y given X .
d) (3 pts) Find E [Y |X = x].
e) (3 pts) Find the value of E [Y ].
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ECEn 370 Final
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14. (10 points) MATLAB Problem. I have a random variable
p X that is uniformly distributed from -2 to 2. Suppose
that I want to estimate the PDF of a random variable Y = |X|. For my computation, I will generate 10,000 trials
of X and then compute the PDF of Y based on those results. Fill in the balnk lines to generate the necessary code
to visualize the PDF of Y .
% S e t s the number o f t r i a l s t h a t w i l l be run
N=
%Generates N outcomes from the uniform d i s t r i b u t i o n from −2 to 2
X=
% Computes the Y outcomes from the X outcomes
Y=
% The f o l l o w i n g code s p l i t s the Y data i n t o 10 b i n s
[ n , xout ] =
% The f o l l o w i n g code p l o t s the data a f t e r a p p r o p r i a t e s c a l i n g
t i t l e ( ' D i s t r i b u t i o n o f random v a r i a b l e Y ' ) ;
x l a b e l ( ' Values o f Y ' ) ;
y l a b e l ( ' Estimated d e n s i t y at Y Values ' ) ;
12
Standard Normal Table, P(Z < z)
.00
.01
0.0
0.5000
0.5040
0.1
0.5398
0.5438
0.2
0.5793
0.5832
0.3
0.6179
0.6217
0.4
0.6554
0.6591
0.5
0.6915
0.6950
0.6
0.7257
0.7291
0.7
0.7580
0.7611
0.8
0.7881
0.7910
0.9
0.8159
0.8186
1.0
0.8413
0.8438
1.1
0.8643
0.8665
1.2
0.8849
0.8869
1.3
0.9032
0.9049
1.4
0.9192
0.9207
1.5
0.9332
0.9345
1.6
0.9452
0.9463
1.7
0.9554
0.9564
1.8
0.9641
0.9649
1.9
0.9713
0.9719
2.0
0.9772
0.9778
2.1
0.9821
0.9826
2.2
0.9861
0.9864
2.3
0.9893
0.9896
2.4
0.9918
0.9920
2.5
0.9938
0.9940
2.6
0.9953
0.9955
2.7
0.9965
0.9966
2.8
0.9974
0.9975
2.9
0.9981
0.9982
3.0
0.9987
0.9987
3.1
0.9990
0.9991
3.2
0.9993
0.9993
3.3
0.9995
0.9995
3.4
0.9997
0.9997
3.5
0.9998
0.9998
3.6
0.9998
0.9998
3.7
0.9999
0.9999
3.8
0.9999
0.9999
.02
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.9998
0.9999
0.9999
0.9999
.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
0.9999