Disc05c

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School
University of California, Santa Barbara **We aren't endorsed by this school
Course
CS 70
Subject
Statistics
Date
Sep 2, 2023
Pages
3
Uploaded by JudgeJellyfishPerson875 on coursehero.com
variance covariance variance: measures how much on average the variable deviates from its expectation Var(x) = EC(X-ECX))2) = ECX2] - ECX]2 remember E(x2) F E(X)E(X) !!! · standard deviation: o = wr(x) · With a constant c, Var(cX) = cVar(x) - constant is squared Var(X+ c) = Var(X) I constant goes away covariance: association between two RVs X and Y COV(X, y) = EC(X-E(X))(y- E(Y))] = ECX47-ECXJECY] remember E(XY) * ECX)ECYS !!! - <O x and y are negatively correlated COV(X, Y): =0 no correlation but does not imply independence E >O X and y are positively correlated general principles for variance and covariance: · COV(X, y) = COV(Y,X) · Cor(X +Y,z) = (or(X,z) + (or(Y,z) · Cor(aX, y) = aCor(X, y) · Var (x) = COV(X, X) · Var (x,y) = Var (x) + Var(Y) + 2 cor(X,x) if X and Y are independent, · COV(X, y) = 0 · ECXY] = E(XJECY] · Var(x+Y) = Var(X) +Var(Y)
CS 70 Discrete Mathematics and Probability Theory Summer 2023 Huang, Suzani, and Tausik DIS 5C 1 Dice Variance Note 16 (a) Let X be a random variable representing the outcome of the roll of one fair 6-sided die. What is Var ( X ) ? (b) Let Z be a random variable representing the average of n rolls of a fair 6-sided die. What is Var ( Z ) ? 2 Elevator Variance Note 16 A building has n upper floors numbered 1 , 2 ,..., n , plus a ground floor G . At the ground floor, m people get on the elevator together, and each person gets off at one of the n upper floors uniformly at random and independently of everyone else. What is the variance of the number of floors the elevator does not stop at? CS 70, Summer 2023, DIS 5C 1 F(x2) = 12.6 + 22.5 + 32.6 + 42.6 + 52.4 + 62.6 = I E(X]2 = 1.5 + 2.6 + 3.6 + 4.6 + 6.6 + 6.6 = E var(x) = E(x2) -E(X)2 = B - (E) = more times to roll the dice, variance Yar(t,Xi)= varl, i becomes lower (better estimate = Xil =2.n. - I will be to true average) from part a N: # of floors elevator does not stop at Var(N) = ECN2] - ECN)2 Ni: Whether no one gets off on floor i N = N, + N2+... + Nn 4 every soft ECN2 = E(, , Ni] = E, E(NiT = E(2")w = n(-)) E(N2S = E[(N,+... + Nn)"7 = E(=,Ni2] +ECE, NiN;] = E(Ni +, ECNiN; E(Ni2] = E(NiT E(NiN;]: both Ni = 1 & Ni = 1 blc if Ni = 1, then N: must be E(NiC] = (*) m F(NiNi)= (* cannot get off E())m = n(t)m on 2 floors var(I) = n(*) " + n(n - x(= 2)r - (n(2))m)- EjE(NiN;T = nen-1)(*) m ChoiCeS - Envices for i for j
3 Covariance Note 16 (a) We have a bag of 5 red and 5 blue balls. We take two balls uniformly at random from the bag without replacement. Let X 1 and X 2 be indicator random variables for the events of the first and second ball being red, respectively. What is cov ( X 1 , X 2 ) ? Recall that cov ( X , Y ) = E [ XY ] - E [ X ] E [ Y ] . (b) Now, we have two bags A and B, with 5 red and 5 blue balls each. Draw a ball uniformly at random from A, record its color, and then place it in B. Then draw a ball uniformly at random from B and record its color. Let X 1 and X 2 be indicator random variables for the events of the first and second draws being red, respectively. What is cov ( X 1 , X 2 ) ? CS 70, Summer 2023, DIS 5C 2 1 ball is red xi3xi: 2o ball is blue p = F = COV(X,, X2) = ECX, Xc] -ECX,3 E(X2] E(X,] = I FCX2) = I by symmetry since we don't know what first ball is E(X,X2) = P(X, = 11X2 = 1) = P(X, = 1]P(Xz = 1(X, = 1) = I = COV(X,X2) = 2 - 5 = - 4 = - 55 Yi3x:: . . e COV(X,, X2) = FCX, X27-ECX,7ECX2] E(X,) = I F(X) = = + I = I by symmetry, 50% chance of - - I redtred blue+red getting + or -1 E(X,Xz) = P(X, = 11X2 = 1) = 5.4 = π covIX,X2) = - It = π - 4 = y
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