Question

1) Give an example for each of the following:

a) A random experiment with finite space of elementary events.

b) A random experiment with infinite countable space of elementary events.

c) A random experiment with continuous space of elementary events.

Answer 1

Sample space S of a random experiment is defined as the set of all
possible outcomes.

A sample space can be finite, countably infinite or uncountably infinite.
1. Toss a coin two times
S1 = {(H, H), (H, T), (T, H), (T, T)}
S1 is countable, S1 is called a discrete sample space. Define
B = {H, T}, then S1 = B × B.

2. Toss a dice until a ‘six’ appears and count the number of
times the dice was tossed.
S2 = {1, 2, 3, …};
S2 is discrete and countably infinite (one-to-one correspondence
with positive integers)
3. Pick a number X at random between zero and one, then pick a
number Y at random between zero and X.
S3 = {(x, y): 0 ≤ y ≤ x ≤ 1};
S3 is a continuous sample space.