Unizor - Probability - Correlation - Problems 6

Notes to a video lecture on http://www.unizor.com


Random Variables
Problems 6 (Correlation)

As always, try to solve any problems presented on this Web site just by yourself and check against the answers provided.
Only then study the suggested solutions.

Problem 6.1.
Consider two random variables,ξ and η, not necessarily independent, each taking two different values as follows:
P{ξ=x₁} = 1/2
P{ξ=x₂} = 1/2
P{η=y₁} = 1/2
P{η=y₂} = 1/2

Assume that
P{ξ=x₁ & η=y₁} = r

Analyze the domain and the range of correlation coefficient R(ξ,η) of these random variables.

Hint:
Use the formula of correlation R(ξ,η) in terms of r from the previous lecture ("Problems 5").

Answer:
Domain:
r should be in the interval
[0,1/2]
Range: [−1,1]

Problem 6.2.
Consider two random variables,ξ and η, not necessarily independent, each taking two different values as follows:
P{ξ=x₁} = 1/2
P{ξ=x₂} = 1/2
P{η=y₁} = 2/3
P{η=y₂} = 1/3

Assume that
P{ξ=x₁ & η=y₁} = r

Analyze the domain and the range of correlation coefficient R(ξ,η) of these random variables.

Answer:
Domain:
r should be in the interval
[1/6,1/2]
Range: [−√2/2,√2/2]

Problem 6.3.
Consider the same two random variables, ξ and η, not necessarily independent, each taking two different values as follows:
P{ξ=x₁} = p
P{ξ=x₂} = 1−p
P{η=y₁} = q
P{η=y₂} = 1−q

Assume that
P{ξ=x₁ & η=y₁} = r
and, for definitiveness, p is not smaller than 1/2 and is not greater than q,
that is
1/2 ≤ p ≤ q
(we can always choose x₁ and y₁ as values with greater or equal probability than, correspondingly, x₂ and y₂)

Analyze the domain and the range of correlation coefficient R(ξ,η) of these random variables.

Answer:
Domain:
r should be in the interval
[p+q−1,p]
Range:
from √[(1−p)(1−q)]/[pq]
to √[p(1−q)]/[q(1−p)]