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Probability Review for Financial Engineers Part 2 1 Conditional Probability The conditional probability that E occurs given that F has occurred is denoted by π(πΈ|πΉ) If P(F) > 0 then π πΈ πΉ = π(πΈπΉ) π(πΉ) 2 Example β 2 dice 2 dice are rolled - a red dice and a green dice, What is the probability distribution for the total? 2 - 1/36 3 - 2 * (1/36) β¦ 7 β 6 * (1/36) β¦ 11 β 2 (1/36) 12 β 1/36 3 2 Dice example continued What is the expected value? 7 Ex) What is the probability distribution function for the total given that the Green dice was a 3, that is P(T|G=3) 4 β 1/6 5 β 1/6 6 β 1/6 7 β 1/6 8 β 1/6 9 β 1/6 4 Example) Playing Cards Selecting a card a standard 52 playing card deck What is the probability of getting an ace? 4/52 = 1/13 What is the probability of getting a ace given that someone already removed a jack from the deck? 4/51 the removal of a jack means that a non-ace has been removed from the deck What is the probability of getting an ace given that someone already removed a spade from the deck? 1/13 the removed cards suit is independent of the rank question. 5 Joint Cumulative Distributions F(a,b) = P(Xβ€a, Yβ€b) The distribution of X can be obtained from the joint distribution of X and Y as follows πΉπ = π π < π = π π < π|π < β = π( lim π < π|π < π ) πββ = lim π π < π|π < π πββ = lim πΉ(π, π) πββ = πΉ(π, β) 6 Example β Time between arrivals A market buy order and a market sell order arrive uniformly distributed between 1 and 2pm. Each person puts a 10 minute time limit on each order. What is the probability that the trade will not be executed because of a timeout? This would be the P(B +10 < S) + P(S+10 < B) = 2 P(B +10 < S) =2 π π, π ππ ππ π΅+10<π =2 ππ΅ (π)ππ (π )ππ ππ π΅+10<π 60 π β10 1 60 =2 10 = 2 60 0 2 ππ ππ 60 2 π β 10 ππ 10 = 25 36 7 Expected Values of Joint Densities Suppose f(x,y) is a joint distribution πΈ[π π β π ] β β = π π₯ β π₯ π(π₯, π¦)ππ₯ ππ¦ ββ ββ β = β π π₯ β π₯ ππ (π₯)ππ (π¦)ππ₯ ππ¦ ββ ββ β = β β π₯ ππ (π¦)ππ¦ ββ π π₯ ππ (π₯)ππ₯ ββ = πΈ[β π ]πΈ[π π ] 8 Covariance of 2 Random Variables πΆππ£ π, π = πΈ (π β πΈ π β πβπΈ π ] = πΈ ππ β πΈ π π β ππΈ π + πΈ π πΈ[π] = πΈ ππ β πΈ π πΈ[π] β πΈ[π]πΈ π + πΈ π πΈ[π] = πΈ ππ β πΈ π πΈ[π] Note that is X and Y are independent, then the covariance = 0 9 Variance of sum of random variables πππ π + π = πΈ[ π + π β πΈ π + π = πΈ = πΈ[ π + π β πΈπ β πΈπ 2 ] = πΈ[ π β πΈπ + π β πΈπ 2 ] π β πΈπ 2 + π β πΈπ 2 2 ] + 2 π β πΈπ π β πΈπ = πΈ π β πΈπ 2 ] + πΈ[ π β πΈπ + 2πΈ[ π β πΈπ π β πΈπ ] 2 πππ π + π = πππ π + πππ π + 2πΆππ£(π, π) 10 Correlation of 2 random variables As long as Var(X) and Var(Y) are both positive, the correlation of X and Y is denotes as π π, π = πΆππ£(π, π) πππ π πππ(π) It can be shown that β1 β€ π π, π β€ 1 The correlation coefficient is a measure of the degree of linearity between X and Y π π, π = 0 means very little linearity π π, π ππππ + 1 means X and Y increase and decrease together π π, π ππππ β 1 means X and Y increase and decrease inversely 11 Central Limit Theorem Loosely put, the sum of a large number of independent random variables has a normal distribution. Let π1 , π2 β¦ be a sequence of independent and identically distributed random variables each having mean π and variance π 2 Then the distribution of π1 + β― + ππ β nπ π π Tends to a standard normal as nο β, that is π1 + β― + ππ β nπ 1 π β€π β π π 2π π π βπ₯ 2 /2 ππ₯ ββ 12