Assignment 1-Probability: basic definition
Due: 17.3.2016. Need to do 3,5,6,7,8. 1,2, 4 are optional.
1. Prove from basic definitin that P(Ac ) = 1−P(A) (use only the three properties defining
a probability measure)
2. Prove by induction on n that
P (∪ni=1 Ai )
=
n
X
P(Ai )−
i=1
X
P(Ai1 ∩Ai2 )+
i1 <i2
X
P(Ai1 ∩Ai2 ∩Ai3 )+· · ·+(−1)n+1 P(A1 ∩· · ·∩An ).
i1 <i2 <i3
3. You toss a fair dice 10 times. What is the probability that ”6” appears at least once?
4. Let Fi , i = 1, 2, · · · be a sequence of σ-algebra defined on the same sample space Ω .
Prove that ∩∞
i=1 Fi is a σ-algebra.
Is ∪∞
i=1 a σ algebra? prove or give a counter example.
5. Let A and B be events. P(A) = 3/4 and P(B) = 1/3 show that 1/12 ≤ P(A∩B) ≤ 1/3
and give example that both extremes are possible. Find corresponding bounds on
P(A ∪ B).
6. Assume that you throw a dice once.
(i) Describe the sample space.
(ii) Find the σ algebra generated by the sets {6}, {1, 3, 5}
7. Let X be a Normal distributed random variable with mean µ and variance σ 2 . Find
E[etX ] . This is the moment generating function of X.
8. For each positive integer n define fn to be the normal density with mean 0 and variance
n., i.e.
x2
1
e− 2n
fn (x) = √
2πn
What is f (x) = limn→∞ fn (x)
R∞
What is limn→inf ty −∞ fn (x)dx?
Is
Z
lim
n→∞
∞
Z
∞
fn (x)dx =
−∞
f (x)dx ?
−∞
Explain why this do violated the monotone convergence Theorem?
9. Let P be the Lebesgue measure on [0, 1], (i.e. the probability space is ([0, 1], B, L)).
Define:
0
if 0 ≤ w < 1/2
Z(w) =
2
if 1/2 ≤ w ≤ 1
For A ∈ B[0, 1] , define
Z
P̃(A) =
Z(w)dP(w)
A
1
(i) Show that P̃ is a probability measure.
(ii) Show that if P(A) = 0, then P̃(A) = 0. We say that P̃is absolutely continuous
with respect to P.
(iii) Show that there is a set A for which P̃(A) = 0 and P(A) > 0. In other words P̃
and P are not equvalent.
2