Massachusetts Institute of Technology
Sec. 3.3
Department of Electrical Engineering & Computer Science
Probabilistic Systems Analysis
Normal6.041/6.431:
Random Variables
155
(Quiz Ϯ | Fall 2009)
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
.5000
.5398
.5793
.6179
.6554
.5040
.5438
.5832
.6217
.6591
.5080
.5478
.5871
.6255
.6628
.5120
.5517
.5910
.6293
.6664
.5160
.5557
.5948
.6331
.6700
.5199
.5596
.5987
.6368
.6736
.5239
.5636
.6026
.6406
.6772
.5279
.5675
.6064
.6443
.6808
.5319
.5714
.6103
.6480
.6844
.5359
.5753
.6141
.6517
.6879
0.5
0.6
0.7
0.8
0.9
.6915
.7257
.7580
.7881
.8159
.6950
.7291
.7611
.7910
.8186
.6985
.7324
.7642
.7939
.8212
.7019
.7357
.7673
.7967
.8238
.7054
.7389
.7704
.7995
.8264
.7088
.7422
.7734
.8023
.8289
.7123
.7454
.7764
.8051
.8315
.7157
.7486
.7794
.8078
.8340
.7190
.7517
.7823
.8106
.8365
.7224
.7549
.7852
.8133
.8389
1.0
1.1
1.2
1.3
1.4
.8413
.8643
.8849
.9032
.9192
.8438
.8665
.8869
.9049
.9207
.8461
.8686
.8888
.9066
.9222
.8485
.8708
.8907
.9082
.9236
.8508
.8729
.8925
.9099
.9251
.8531
.8749
.8944
.9115
.9265
.8554
.8770
.8962
.9131
.9279
.8577
.8790
.8980
.9147
.9292
.8599
.8810
.8997
.9162
.9306
.8621
.8830
.9015
.9177
.9319
1.5
1.6
1.7
1.8
1.9
.9332
.9452
.9554
.9641
.9713
.9345
.9463
.9564
.9649
.9719
.9357
.9474
.9573
.9656
.9726
.9370
.9484
.9582
.9664
.9732
.9382
.9495
.9591
.9671
.9738
.9394
.9505
.9599
.9678
.9744
.9406
.9515
.9608
.9686
.9750
.9418
.9525
.9616
.9693
.9756
.9429
.9535
.9625
.9699
.9761
.9441
.9545
.9633
.9706
.9767
.9772 normal
.9778 .9783
.9788
.9817
The2.0standard
table.
The .9793
entries.9798
in this.9803
table .9808
provide.9812
the numeri­
.9826
.9830
.9834
.9838
.9846 normal
.9850 .9854
cal 2.1
values.9821
of Φ(y)
= P(Y
≤ y),
where
Y is.9842
a standard
random.9857
vari­
able,2.2for .9861
y between
example,
find Φ(1.71),
we look
.98640 and
.98681.99.
.9871For
.9875
.9878 to.9881
.9884 .9887
.9890at
the 2.3
row corresponding
1.7 and
column
corresponding
0.01, so.9916
that
.9893 .9896 to
.9898
.9901the.9904
.9906
.9909 .9911to .9913
Φ(1.71)
.9564. .9920
When .9922
y is negative,
the value
of Φ(y)
be found
the
2.4 =.9918
.9925 .9927
.9929
.9931can.9932
.9934using
.9936
formula Φ(y) = 1 − Φ(−y).
2.5
2.6
2.7
2.8
2.9
.9938
.9953
.9965
.9974
.9981
.9940
.9955
.9966
.9975
.9982
.9941
.9956
.9967
.9976
.9982
.9943
.9957
.9968
.9977
.9983
.9945
.9959
.9969
.9977
.9984
.9946
.9960
.9970
.9978
.9984
.9948
.9961
.9971
.9979
.9985
.9949
.9962
.9972
.9979
.9985
.9951
.9963
.9973
.9980
.9986
.9952
.9964
.9974
.9981
.9986
3.0
3.1
3.2
3.3
3.4
.9987
.9990
.9993
.9995
.9997
.9987
.9991
.9993
.9995
.9997
.9987
.9991
.9994
.9995
.9997
.9988
.9991
.9994
.9996
.9997
.9988
.9992
.9994
.9996
.9997
.9989
.9992
.9994
.9996
.9997
.9989
.9992
.9994
.9996
.9997
.9989
.9992
.9995
.9996
.9997
.9990
.9993
.9995
.9996
.9997
.9990
.9993
.9995
.9997
.9998
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
Problem 2. (42 points)
The random variable X is exponential with parameter 1. Given the value x of X, the random variable Y is
exponential with parameter equal to x (and mean 1/x).
Note: Some useful integrals, for λ > 0:
∞
xe−λx dx =
0
∞
1
,
λ2
x2 e−λx dx =
0
2
.
λ3
(a) (7 points) Find the joint PDF of X and Y .
(b) (7 points) Find the marginal PDF of Y .
(c) (7 points) Find the conditional PDF of X, given that Y = 2.
(d) (7 points) Find the conditional expectation of X, given that Y = 2.
(e) (7 points) Find the conditional PDF of Y , given that X = 2 and Y ≥ 3.
(f) (7 points) Find the PDF of e2X .
Problem 3. (10 points)
For the following questions, mark the correct answer. If you get it right, you receive 5 points for that question.
You receive no credit if you get it wrong. A justification is not required and will not be taken into account.
Let X and Y be continuous random variables. Let N be a discrete random variable.
(a) (5 points) The quantity E[X | Y ] is always:
(i) A number.
(ii) A discrete random variable.
(iii) A continuous random variable.
(iv) Not enough information to choose between (i)-(iii).
(b) (5 points) The quantity E[ E[X | Y, N ] | N ] is always:
(i) A number.
(ii) A discrete random variable.
(iii) A continuous random variable.
(iv) Not enough information to choose between (i)-(iii).
Problem 4. (25 points)
The probability of obtaining heads in a single flip of a certain coin is itself a random variable, denoted by Q,
which is uniformly distributed in [0, 1]. Let X = 1 if the coin flip results in heads, and X = 0 if the coin flip
results in tails.
(a)
(i) (5 points) Find the mean of X.
(ii) (5 points) Find the variance of X.
(b) (7 points) Find the covariance of X and Q.
(c) (8 points) Find the conditional PDF of Q given that X = 1.
3
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
Problem 5. (21 points)
Let X and Y be independent continuous random variables with marginal PDFs fX and fY , and marginal
CDFs FX and FY , respectively. Let
S = min{X, Y },
L = max{X, Y }.
(a) (7 points) If X and Y are standard normal, find the probability that S ≥ 1.
(b) (7 points) Fix some s and £ with s ≤ £. Give a formula for
P(s ≤ S and L ≤ £)
involving FX and FY , and no integrals.
(c) (7 points) Assume that s ≤ s + δ ≤ £. Give a formula for
P(s ≤ S ≤ s + δ, £ ≤ L ≤ £ + δ),
as an integral involving fX and fY .
Each question is repeated in the following pages. Please write your answer on
the appropriate page.
4
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
Problem 2. (42 points)
The random variable X is exponential with parameter 1. Given the value x of X, the random variable Y is
exponential with parameter equal to x (and mean 1/x).
Note: Some useful integrals, for λ > 0:
Z ∞
Z ∞
2
1
x2 e−λx dx = 3 .
xe−λx dx = 2 ,
λ
λ
0
0
(a) (7 points) Find the joint PDF of X and Y .
(b) (7 points) Find the marginal PDF of Y .
5
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
(c) (7 points) Find the conditional PDF of X, given that Y = 2.
(d) (7 points) Find the conditional expectation of X, given that Y = 2.
6
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
(e) (7 points) Find the conditional PDF of Y , given that X = 2 and Y ≥ 3.
(f) (7 points) Find the PDF of e2X .
7
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
Problem 3. (10 points)
For the following questions, mark the correct answer. If you get it right, you receive 5 points for that question.
You receive no credit if you get it wrong. A justification is not required and will not be taken into account.
Let X and Y be continuous random variables. Let N be a discrete random variable.
(a) (5 points) The quantity E[X | Y ] is always:
(i) A number.
(ii) A discrete random variable.
(iii) A continuous random variable.
(iv) Not enough information to choose between (i)-(iii).
(b) (5 points) The quantity E[ E[X | Y, N ] | N ] is always:
(i) A number.
(ii) A discrete random variable.
(iii) A continuous random variable.
(iv) Not enough information to choose between (i)-(iii).
Problem 4. (25 points)
The probability of obtaining heads in a single flip of a certain coin is itself a random variable, denoted by Q,
which is uniformly distributed in [0, 1]. Let X = 1 if the coin flip results in heads, and X = 0 if the coin flip
results in tails.
(a)
(i) (5 points) Find the mean of X.
8
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
(ii) (5 points) Find the variance of X.
(b) (7 points) Find the covariance of X and Q.
9
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
(c) (8 points) Find the conditional PDF of Q given that X = 1.
10
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
Problem 5. (21 points)
Let X and Y be independent continuous random variables with marginal PDFs fX and fY , and marginal
CDFs FX and FY , respectively. Let
S = min{X, Y },
L = max{X, Y }.
(a) (7 points) If X and Y are standard normal, find the probability that S ≥ 1.
11
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Quiz Ϯ | Fall 2009)
(b) (7 points) Fix some s and £ with s ≤ £. Give a formula for
P(s ≤ S and L ≤ £)
involving FX and FY , and no integrals.
(c) (7 points) Assume that s ≤ s + δ ≤ £. Give a formula for
P(s ≤ S ≤ s + δ, £ ≤ L ≤ £ + δ),
as an integral involving fX and fY .
12
MIT OpenCourseWare
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6.041SC Probabilistic Systems Analysis and Applied Probability
Fall 2013
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