EE5137R 2012/13 Problem Set 3
1. (This problem is based on an example presented in class.) Consider a two-step
experiment where we first roll a fair die, call the result X, and then if X = x, we
pick a number uniformly from the set cfw_1, 2, . . . , x, an
EE5137R 2012/13 Problem Set 3 Solutions
Correction of PS2 solutions:
In Problem 9(b), it should be P [Y = 1, X = 2] = P [3 ] =
Then use the definition of conditional probability,
P [Y = 1 | X = 2] =
instead of P [Y = 1 | X = 2].
P [3 ]
P [Y = 1, X =
EE5137R 2012/13 Problem Set 4 Solutions
1. Problem 3.13
V ar [X] = E X 2 (E [X])2 =
k 2 q k1 p
kq k1 p
Rewrite (1) as
EE5137R 2012/13 Problem Set 2 Solutions
1. Problem 2.12
According to Axiom III,
if A B = ,
then P (A B) = P (A) + P (B)
Therefore, (2.22) is true for M = 2.
Assume (2.22) holds for M = N,
(N 2). i.e.
if i, j cfw_0, 1, . . . , N and i 6= j, Ai Aj = ,
EE5137R 2012/13 Problem Set 5 Solutions
1. Problem 4.3
(a) P (Z z | Z = Xi ) = FXi (z), P (Z = Xi ) = pi P (Z z, Z = Xi ) =
pi FXi (z)
The event cfw_Z z = ni=1 cfw_Z z, Z = Xi
Therefore, FZ (z) = ni=1 pi FXi (z)
(b) E(Z) = E(E(Z | Z = Xi ) = E(E(Xi )
EE5137R 2012/13 Problem Set 5
1. Problem 4.3.
2. Problem 4.4 Part (b) is for more advanced students. Please look up the RiemannStieltjes integral on Wikipedia.
3. Let X be a continuous random variable with range (a, b] and the PDF fX (x).
(a) Consider the
EE5137R 2012/13 Problem Set 4
1. Problem 3.13.
2. Problem 3.14.
3. Problem 3.15a, b, c, e.
4. Problem 3.16a.
5. Suppose that on average, 1 in 5 visitors to a clinic have a serious medical problem.
Of those with serious conditions, 40 percent are referred
EE5137R 2012/13 Problem Set 2
1. Problem 2.12. (Google mathematical induction and pick the first link, if you dont
know the proof by induction technique.)
2. Prove from first principles the Chain Rule for a sequence of n events A1 , . . . , An :
P (An A1
EE5137R 2012/13 Problem Set 1 Solutions
1. Problem 2.1(a,b).
(a) = cfw_HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T
(b) E0 = cfw_T T T , E1 = cfw_HT T, T HT, T T H, E2 = cfw_HHT, HT H, T HH, E3 =
Ei contains Ni sample points: N0 = 1, N1 = 3, N2
EE5137R 2012/13 Problem Set 1
1. Problem 2.1(a,b).
2. Problem 2.3.
3. Problem 2.4.
4. Problem 2.5.
5. Problem 2.7.
6. Problem 2.18.
7. Problem 2.19.
8. A coin is tossed twice and the sides facing up recorded. (a) Write down the sample
space of this experi