4.1 Probabilities

Question 1

What is a sample space (omega symbol)

Answer 1

Set of elementary outcomes of experiment

Question 2

An experiment is characterised by a permutation of 1…n.

Sn (sample space) is the set of all permutations. What is the size of Sn?

Answer 2

n!

Question 3

Let A: Sn –> {1, … ,n} be a random variable which measures something in an experiment.

Example: A((4,3,6,7,1,2,4,6,5)) = 5

What are these different elements:
A
(4,3,6,7,1,2,4,6,5)
5

Answer 3

A = random variable
(4,3,6,7,1,2,4,6,5) = outcome
5 = observation

Question 4

What is the formula for the expected value of A? And explain what it means

Answer 4

For every value in the set of observations, we take the observation and multiply it by the probability of that observation

Question 5

What is an indicator random variable

Answer 5

A random variable that only ever outputs true/false via a 1/0

Question 6

if there are 5 observations, what is the probability that the 5th one is the highest

Answer 6

(5 -1)! / (5)!

24/120 = 1 / 5

(i-1)! / i! = 1/i

Question 7

If there are 8 observations (1 … 8), what is the probability that the i’th term is greater than average?

Answer 7

Number of permutations where i’th > n/2
/
Number of possible permutations

=

(n-1)! * n/2
/
n!

=

(7! * 4) / 8! = 0 .5