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