Probability Concept

Probability Concept

1. Definition

   • Event: a specific set of outcomes
   • An Outcome: a possible value of a random variable
   • Random variable: a quantity whose future outcomes are uncertain
   • Probability P(E) or the Odds
     • Mutually exclusive and exhaustive event: 1
     • Subjective Probability
     • Objective Probability
       • Empirical Prob: Past Data
       • Prior Prob: Deductive

2. The Odds for E: Prob will occur/ Not occur

3. Conditional and Joint Prob

   • Unconditional Prob: Without any restriction
   • Conditional Prob: Given that another event has occured
   • P(AB) - The Prob of A and B: P(AB) = P(A)P(B/A)
   • P(A/B): The Prob of A given B

4. Addition Rule: P(A or B)=P(A) + P(B) - P(AB)

5. Independent: The incurrence of one event has no influence on the occurrence of another event: P(A/B) = P(A)

   • Multiplication Rule: P(AB)=P(A) P(B)

6. Total Prob Rule: P(A)= P(A/B).P(B)+...P(A/Bn).P(Bn) with P(B) + ... + P(Bn) = 1

7. Expected Value: Weighted average E(X)=P(X1).X1+...+P(Xn).Xn

   • Variance and standard deviation measure the dispersion of outcomes around the expected value or forecast: σ^2 (X) = E {[X − E (X) ]^2}
   • E(X) = E(X | S1)P(S1)+...+ E(X | Sn)P(Sn) where P(S1)+...+P(Sn)=1
     • Conditional E(X | S) = P(X1 | S)X1 + P(X2 | S)X2 + … + P(Xn | S)Xn

8. PORTFOLIO

   • Expected Return: E (Rp) = w1 E (R1) + w2 E (R2) + … + wn E (Rn)
   • σ 2 (Rp) = w1^2 σ 2 (R1) + w2^2 σ 2 (R2) + + 2 w1 w2 Cov (R1, R2) Cov (R1, R2) = E [ (R1 − E R1) (R2 − E R2) ]
   • Corr(RiRj) = Cov(Ri,Rj)/[σ(Ri)σ(Rj)]

9. LABELING

   • The number of ways that n objects can be labeled with k different labels, with n1 of the first type...n !(n 1!*n2!… nk !)
   • Choose r objects from a total of n objects, when the order in does or doesn't matter
     • Combination Formula: nCr = n!/ (r!(n-r)!)
     • Permutation Formula: nPr = n!/ (n-r)!

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