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

__Random Variables__

Problems 6 (Correlation)

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=x_{1}} = 1/2

**P**{ξ=x_{2}} = 1/2

**P**{η=y_{1}} = 1/2

**P**{η=y_{2}} = 1/2Assume that

**P**{ξ=x_{1}& η=y_{1}} = rAnalyze the domain and the range of

*correlation coefficient*

*of these random variables.*

**R**(ξ,η)*Hint*:

Use the formula of

*correlation*

*in terms of*

**R**(ξ,η)*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=x_{1}} = 1/2

**P**{ξ=x_{2}} = 1/2

**P**{η=y_{1}} = 2/3

**P**{η=y_{2}} = 1/3Assume that

**P**{ξ=x_{1}& η=y_{1}} = rAnalyze the domain and the range of

*correlation coefficient*

*of these random variables.*

**R**(ξ,η)*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=x_{1}} = p

**P**{ξ=x_{2}} = 1−p

**P**{η=y_{1}} = q

**P**{η=y_{2}} = 1−qAssume that

**P**{ξ=x_{1}& η=y_{1}} = rand, 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

*x*and

_{1}*y*as values with greater or equal probability than, correspondingly,

_{1}*x*and

_{2}*y*)

_{2}Analyze the domain and the range of

*correlation coefficient*

*of these random variables.*

**R**(ξ,η)*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)]
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