# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Random variables on a finite sample space: range, preimage events and point probabilities](/en/derivations/derivation-mko-random-variable-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-random-variable)

S={1,2,3,4}, event family F=𝒫(S), weights w₁=1/2, w₂=1/4, w₃=1/8, w₄=1/8. The random variable X:S→ℝ is given by the table X(1)=1, X(2)=0, X(3)=1, X(4)=4 (that is, X(s)=(s−2)²). Find the range of X, the preimage events {X=1}, {X≥1} and {X∈{2,3}}, and the point probability P(X=t) of each value, and prove that these point probabilities sum to 1.

Real random variables on finite sample spaces

derivation-mko-random-variable-01



Concept guide



## [Independence of random variables: checking the complete finite joint table, with zero covariance only a consequence](/en/derivations/derivation-mko-random-variable-independence-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-random-variable)

(1) Joint table J₁: X∈{0,1}, Y∈{0,1,2}, P(0,0)=1/12, P(0,1)=1/12, P(0,2)=1/6, P(1,0)=1/6, P(1,1)=1/6, P(1,2)=1/3. Aggregate both marginals from the joint table, check the product equation at all 6 value pairs, and decide whether X and Y are independent; also prove that independence implies independence of any coordinate events {X∈S} and {Y∈T} and E[XY]=E[X]E[Y]. (2) Joint table J₂: X uniform on {−1,0,1} and Y=X². Prove Cov(X,Y)=0 although X and Y are not independent. (3) Use joint table J₃ to show that checking a single value pair is not enough.

Independence of random variables

derivation-mko-random-variable-independence-01





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