Unizor - Random Variables - Problems 4

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


Random Variables
Problems 4 

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 4.1.
Consider a random variable ξand η taking two different values as follows:
P{ξ=x₁} = p
P{ξ=x₂} = 1−p

What is its mathematical expectation and variance?

Answer:
E(ξ)=x₁·p+x₂·(1−p)
Var(ξ)=(x₁−x₂)²·p·(1−p)

Problem 4.2.
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 are the probabilities of all other combinations of values for these random variables, namely:
P{ξ=x₁ & η=y₂} = ?
P{ξ=x₂ & η=y₁} = ?
P{ξ=x₂ & η=y₂} = ?

Answer
P{ξ=x₁ & η=y₂} = p−r
P{ξ=x₂ & η=y₁} = q−r
P{ξ=x₂ & η=y₂} = 1+r−p−q

Problem 4.3.
Consider the same two random variables, ξ and η, as described in the previous problem.
What is the mathematical expectation of their product?

Answer:
E(ξ·η) =
= x₁·y₁·r +
+ x₁·y₂·(p−r) +
+ x₂·y₁·(q−r) +
+ x₂·y₂·(1+r−p−q)