## Monday, July 11, 2016

### 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?

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} = ?

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?