6.041/6.431 Fall 2009 Quiz 1
Tuesday, October 13, 12:05 - 12:55 PM.
DO NOT TURN THIS PAGE OVER UNTIL
YOU ARE TOLD TO DO SO
Question Score Out of
A 2
B.1 10
B.2 (a) 10
Name:
B.2 (b i) 12
B.2 (b ii) 12
Recitation Instructor: B.2 (c) 10
B.3 (a) 10
B.3 (b) 12
TA:
B.3 (c) 12
B.3 (d i) 5
B.3 (d ii) 5
Your Grade 100
• This quiz has 2 problems, worth a total of 100 points.
• You may tear apart pages 3 and 4, as per your convenience.
• Write your solutions in this quiz booklet, only solutions in this quiz booklet will be graded.
Be neat! You will not get credit if we can’t read it.
• You are allowed one two-sided, handwritten, 8.5 by 11 formula sheet. Calculators are not
allowed.
• Parts B.2 and B.3 can be done independently.
• You may give an answer in the form of an arithmetic expression (sums, products, ratios,
��
factorials) of numbers that could be evaluated using a calculator. Expressions like 83 or
�5 k
k=0 (1/2) are also fine.
• You have 50 minutes to complete the quiz.
• Graded quizzes will be returned in recitation on Thursday 10/15.
, a linear function, Y = aX + b, the means and the variances of X and Y are
related by
E[Y ] = aE[X] + b, var(Y ) = a2 var(X).
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
We also discussed several special
6.041/6.431: random variables,
Probabilistic and derived their PMF,
Systems Analysis
mean, and variance, as summarized (Fall
in the2009)
table that follows.
Summary of Results for Special Random Variables
Discrete Uniform over [a, b]:
! 1
, if k = a, a + 1, . . . , b,
pX (k) = b−a+1
0, otherwise,
a+b (b − a)(b − a + 2)
E[X] = , var(X) = .
2 12
Bernoulli with Parameter p: (Describes the success or failure in a single
trial.) "
p, if k = 1,
pX (k) =
1 − p, if k = 0,
E[X] = p, var(X) = p(1 − p).
Binomial with Parameters p and n: (Describes the number of successes
in n independent Bernoulli trials.)
# $
n k
pX (k) = p (1 − p)n−k , k = 0, 1, . . . , n,
k
E[X] = np, var(X) = np(1 − p).
Geometric with Parameter p: (Describes the number of trials until the
first success, in a sequence of independent Bernoulli trials.)
pX (k) = (1 − p)k−1 p, k = 1, 2, . . . ,
1 1−p
E[X] = , var(X) = .
p p2
Page 3 of 11
Tuesday, October 13, 12:05 - 12:55 PM.
DO NOT TURN THIS PAGE OVER UNTIL
YOU ARE TOLD TO DO SO
Question Score Out of
A 2
B.1 10
B.2 (a) 10
Name:
B.2 (b i) 12
B.2 (b ii) 12
Recitation Instructor: B.2 (c) 10
B.3 (a) 10
B.3 (b) 12
TA:
B.3 (c) 12
B.3 (d i) 5
B.3 (d ii) 5
Your Grade 100
• This quiz has 2 problems, worth a total of 100 points.
• You may tear apart pages 3 and 4, as per your convenience.
• Write your solutions in this quiz booklet, only solutions in this quiz booklet will be graded.
Be neat! You will not get credit if we can’t read it.
• You are allowed one two-sided, handwritten, 8.5 by 11 formula sheet. Calculators are not
allowed.
• Parts B.2 and B.3 can be done independently.
• You may give an answer in the form of an arithmetic expression (sums, products, ratios,
��
factorials) of numbers that could be evaluated using a calculator. Expressions like 83 or
�5 k
k=0 (1/2) are also fine.
• You have 50 minutes to complete the quiz.
• Graded quizzes will be returned in recitation on Thursday 10/15.
, a linear function, Y = aX + b, the means and the variances of X and Y are
related by
E[Y ] = aE[X] + b, var(Y ) = a2 var(X).
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
We also discussed several special
6.041/6.431: random variables,
Probabilistic and derived their PMF,
Systems Analysis
mean, and variance, as summarized (Fall
in the2009)
table that follows.
Summary of Results for Special Random Variables
Discrete Uniform over [a, b]:
! 1
, if k = a, a + 1, . . . , b,
pX (k) = b−a+1
0, otherwise,
a+b (b − a)(b − a + 2)
E[X] = , var(X) = .
2 12
Bernoulli with Parameter p: (Describes the success or failure in a single
trial.) "
p, if k = 1,
pX (k) =
1 − p, if k = 0,
E[X] = p, var(X) = p(1 − p).
Binomial with Parameters p and n: (Describes the number of successes
in n independent Bernoulli trials.)
# $
n k
pX (k) = p (1 − p)n−k , k = 0, 1, . . . , n,
k
E[X] = np, var(X) = np(1 − p).
Geometric with Parameter p: (Describes the number of trials until the
first success, in a sequence of independent Bernoulli trials.)
pX (k) = (1 − p)k−1 p, k = 1, 2, . . . ,
1 1−p
E[X] = , var(X) = .
p p2
Page 3 of 11
