Question

In an experiment of tossing a fair coin four times. Let the sample space Ω be the

number of tails observed and ϕ be the impossible event.

(a) List the Ω and Sigma field F, with the maximum cardinality.

(b) If A1, A2, A3, A4 are subsets of Ω, show that the class of sets F = {ϕ, A1, A2, A3, A4, Ω}

is a σ − f ield.

(c) If P is a function defined on F, what properties must P satisfy for the triple

(Ω, F, P) to be called a probability space.

Answer 1

(a) In an experiment of tossing a fair coin four times the number of tails observed can be any number from $\Omega=\{0,1,2,3,4\}$, which is, hence, the sample space.

The sigma field F with the maximum cardinality, is the set of all subsets of $\Omega$:

F={$\emptyset$, {0}, {1}, {2}, {3}, {4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {0, 1, 2}, {0, 1, 3}, {0, 1, 4}, {0, 2, 3}, {0, 2, 4}, {0, 3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}, {0, 1, 2, 3, 4}}

(b) If A1, A2, A3, A4 are subsets of Ω, show that the class of sets F = {ϕ, A1, A2, A3, A4, Ω}

is a σ−field.

This statement is not true, because the cardinality of any finite σ−field must be $2^n$ for some $n$, which is not the case, since the cardinality of $F$ is 6.

(c) In order to the triple (Ω, F, P) to be called a probability space P must satisfy the following properties:

1) For any $A\in F$ the value of $P(A)$ must be non-negative: $P(A)\geq0$.

2) For any disjoint $A, B\in F$ $P(A\cup B)=P(A)+P(B)$.

3) $P(\Omega)=1$.