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CS70 Guide
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Discrete Math
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Propositional Logic
Proofs
Stable Matching
Graphs
Modular Arithmetic
RSA Cryptography
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Probability
Overview
Counting
Discrete Probability
Discrete Random Variables
Hashing and the Union Bound
Expectation and Variance
Conditional Probability and Bayes' Rule
Concentration Inequalities
Continuous Probability
Markov Chains
Extra Probability
Linear Transformations
Conditional Expectation and Variance
The Beta Family
The Gamma Family
Multivariate Normal
Prediction and Regression
Sums
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Conditional Expectation and Variance
Properties:

Conditional Expectation

​
E(X∣Y)E(X|Y)
is the conditional expectation of
XX
given
YY
​
  • ​
    E(X∣Y=y)E(X|Y=y)
    is a fixed value, but
    E(X∣Y)E(X|Y)
    is a random variable (it is a function of
    YY
    )
  • Iterated expectation:
    E(E(X∣Y))=E(X)E(E(X|Y)) = E(X)
    ​
  • Additivity:
    E(Y+Z∣X)=E(Y∣X)+E(Z∣X)E(Y+Z | X) = E(Y|X) + E(Z|X)
    ​
    • does not work on the right hand side:
      E(Y∣X+Z)≠E(Y∣X)+E(Y∣Z)E(Y | X+Z) \ne E(Y|X) + E(Y|Z)
      ​
  • Linearity:
    E(aX+b∣Y)=aE(X∣Y)+bE(aX + b | Y) = aE(X|Y) + b
    ​
  • Conditioning on the same variable:
    E(g(S)T∣S)=g(S)E(T∣S)E(g(S)T | S) = g(S)E(T|S)
    ​

Conditional Variance

If
Var(Y)Var(Y)
is difficult to find directly, we can use the variance decomposition to condition the variance on another variable.
Var(Y)=E(Var(Y∣X))+Var(E(Y∣X))Var(Y) = E(Var(Y|X)) + Var(E(Y|X))
​
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Conditional Expectation
Conditional Variance