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

__Random Variables__

Problems 4

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

*ξ*and

*η*taking two different values as follows:

**P**{ξ=x_{1}} = p

**P**{ξ=x_{2}} = 1−pWhat is its mathematical expectation and variance?

*Answer:*

**E**(ξ)=x_{1}·p+x_{2}·(1−p)

**Var**(ξ)=(x_{1}−x_{2})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=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}} = rWhat are the probabilities of all other combinations of values for these random variables, namely:

**P**{ξ=x_{1}& η=y_{2}} = ?

**P**{ξ=x_{2}& η=y_{1}} = ?

**P**{ξ=x_{2}& η=y_{2}} = ?*Answer*

**P**{ξ=x_{1}& η=y_{2}} = p−r

**P**{ξ=x_{2}& η=y_{1}} = q−r

**P**{ξ=x_{2}& η=y_{2}} = 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:*

= x

+ x

+ x

+ x

**E**(ξ·η) == x

_{1}·y_{1}·r ++ x

_{1}·y_{2}·(p−r) ++ x

_{2}·y_{1}·(q−r) ++ x

_{2}·y_{2}·(1+r−p−q)
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