Unizor - Random Variables - Problems 5

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


Random Variables
Problems 5 (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 5.1.
Consider 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

What is the covariance of these random variables?

Answer
(x₁−x₂)·(y₁−y₂)·(r−pq)

Problem 5.2.
For the same variables as in a previous problem calculate theircorrelation coefficient

Answer
K(r−pq)/√p(1−p)q(1−q)
where K equals to +1 or −1 depending on a sign of an expression
(x₁−x₂)·(y₁−y₂)
that participates in a covariance of these random variables.