ece 6001
ECE 6001, Probability and Random Variables
OSU, Autumn 2012
Aug. 28, 2012
Due: Sep. 4, 2012
Problem Set 1
Problem 1
For the set Ω = {1, 2, 3, 4},
(a) Find the minimal field that contains the subsets {1, 2} and {2, 4}.
(b) Find the minimal field that contains the subsets {1, 2} and {3, 4}.
Problem 2
Consider the random experiment: The time until a PC fails is observed.
(a) Define a sample space Ω for this experiment.
(b) Describe a possible relevant choice for the field F .
(c) Define two events that are mutually exclusive.
(d) Define two events that have a nonempty intersection.
Problem 3
A photon counter connected to the output of a fiber detects the number of photons,
{Ni , i 1}, received for successive pulses generated by a laser connected to the input of the fiber. Specify which one of the following sequences of events, {Ek , k 1} is increasing, decreasing or none. Very briefly explain why.
(a) Ek = {N1
k} for k
1
(b) Ek = {Nk = 0} for k
1
(c) Ek = {minm
k
Nm = 0} for k
1
(d) Ek = {maxm
k
Nm
1
k} for k
Problem 4
For a sequence of events, {En , n
1}, prove the following:
(a) The union bound,
∞
∞
Ei
P
P (Ei )
i=1
i=1
1
(b) If En ր (increasing), then lim P (En ) = P
n→∞
lim En
n→∞
c
Also using the fact that En ց, argue that the same is true for decreasing sequences.
Hint: In both parts construct another sequence, {Fn , n for which ∞ Fi = ∞ Ei . i=1 i=1
1}, of mutually exclusive events
Problem 5
1
Let E1 and E2 be two events and P (E1 ) = 4 . Evaluate P (E2 ), if c (a) E2 = E1
(b) E1 and E2 are mutually exclusive and P (E1 ∪ E2 ) =
1
2
(c) E1 and E2 are both mutually exclusive and independent
(d) E1 and E2 are independent and P (E1 ∪ E2 ) =
(e) P (E1 |E2 ) =
1
2
and P (E2 |E1 ) =
1
2
3
4
Problem 6 (Optional) Matlab Experiment: Relative Frequency
(a) Write a Matlab program to generate N trials of a random number
OSU, Autumn 2012
Aug. 28, 2012
Due: Sep. 4, 2012
Problem Set 1
Problem 1
For the set Ω = {1, 2, 3, 4},
(a) Find the minimal field that contains the subsets {1, 2} and {2, 4}.
(b) Find the minimal field that contains the subsets {1, 2} and {3, 4}.
Problem 2
Consider the random experiment: The time until a PC fails is observed.
(a) Define a sample space Ω for this experiment.
(b) Describe a possible relevant choice for the field F .
(c) Define two events that are mutually exclusive.
(d) Define two events that have a nonempty intersection.
Problem 3
A photon counter connected to the output of a fiber detects the number of photons,
{Ni , i 1}, received for successive pulses generated by a laser connected to the input of the fiber. Specify which one of the following sequences of events, {Ek , k 1} is increasing, decreasing or none. Very briefly explain why.
(a) Ek = {N1
k} for k
1
(b) Ek = {Nk = 0} for k
1
(c) Ek = {minm
k
Nm = 0} for k
1
(d) Ek = {maxm
k
Nm
1
k} for k
Problem 4
For a sequence of events, {En , n
1}, prove the following:
(a) The union bound,
∞
∞
Ei
P
P (Ei )
i=1
i=1
1
(b) If En ր (increasing), then lim P (En ) = P
n→∞
lim En
n→∞
c
Also using the fact that En ց, argue that the same is true for decreasing sequences.
Hint: In both parts construct another sequence, {Fn , n for which ∞ Fi = ∞ Ei . i=1 i=1
1}, of mutually exclusive events
Problem 5
1
Let E1 and E2 be two events and P (E1 ) = 4 . Evaluate P (E2 ), if c (a) E2 = E1
(b) E1 and E2 are mutually exclusive and P (E1 ∪ E2 ) =
1
2
(c) E1 and E2 are both mutually exclusive and independent
(d) E1 and E2 are independent and P (E1 ∪ E2 ) =
(e) P (E1 |E2 ) =
1
2
and P (E2 |E1 ) =
1
2
3
4
Problem 6 (Optional) Matlab Experiment: Relative Frequency
(a) Write a Matlab program to generate N trials of a random number