13.7.26 1 Summer 2013 Stat 100/Sanchez-Intro Probability 1 Continuing last day: Covariance between two random variables. The multinomial distribution. New: Joint continuous densities Juana Sanchez UCLA Dept of Statistics Announcements Stat 100/Sanchez-Intro Probability 2 • Homework 5 due today, 7/30, at the beginning of lecture. • Quiz 4: first 10 minutes of class. • Midterm 2: Thursday 10-11:50. Broad,2160E. Scientific calculator only. Cheat sheet, two sides of 11x8. You must read the instructions in the midterm 2 folder. Only formulas and definitions in the cheat sheet, as in midterm 1. Summer 2013 Show: Var(X+Y)=Var(X)+Var(Y) + 2Cov(X,Y) Summer 2013 Stat 100/Sanchez-Intro Probability 3 In example followed in last Lecture, var(X) =0.3044 Var(Y)=0.56 Cov(X,Y)=0.172 Var(X+Y)= 1.2084 € Var(X + Y ) = E [(X + Y ) − E( X + Y)]^2( )= (X + Y − E(X + Y ))2P( X,Y)y∑x∑= ((X − E(X)) + (Y − E (Y)))2P(X,Y)y∑x∑= [(X − E(X ))2P( X,Y) + ((Y − E(Y))2P( X,Y)y∑x∑+2(X − E(X))(Y − E (Y))P(X,Y )= (X − E(X))2P( X,Y) +y∑x∑(Y − E(Y ))2P( X,Y) +y∑x∑2 (X − E(X))(Y − E (Y))P(X,Y )y∑x∑= Var(X ) + Var (Y) + 2Cov(X,Y )13.7.26 2 Summer 2013 Stat 100/Sanchez-Intro Probability 4 E(X+Y)= 0.16+0.6+0.9+0.8=2.46 € Var(X + Y ) = (0 − 2.46)20.04 + (1− 2.46)20.16 + (2 − 2.46)20.3 + (3 − 2.46)20.3 + (4 − 2.46)22.46 = 1.2084 Relations between covariances Stat 100/Sanchez-Intro Probability 5 € Result 1Since E(a + X) = a +E(X),Cov(a + X,Y ) = E a + X − E (a + X )[ ]Y − E (Y )[ ]{ }= E a + X − a − E (X )[ ]Y − E (Y )[ ]{ }= Cov(X,Y )Result 2Since E(aX) = aE(X), Cov(aX,bY) = E aX - E(aX)[ ]bY − E(bY )[ ]{ }= E ab X - E(X)[ ]Y − E (Y )[ ]{ } = abE X - E(X)[ ]Y − E(Y )[ ]{ }= abCov(X,Y )Summer 2013 Stat 100/Sanchez-Intro Probability 6 € Result 3 (Since if X, Y r..v, E(aX +bY) = aE(X) +bE(Y), Cov aW + bX,cY + dZ( )= E aW + bX − aE (W ) − bE (X ))( )cY + dZ − cE(Y ) − dE (Z)( )[ ]= E a(W − E (W ) + b(X − E (X ))( )c Y − E(Y )( )+ d Z − E(Z )( )( )[ ]= E a(W − E(W ))c Y − E (Y )( )[ ]+ E a(W − E(W ))d Z − E(Z)( )[ ]+E b(X − E (X ))c Y − E (Y)( )[ ]+ E b(X − E (X ))d Z − E (Z)( )[ ]= acE (W − E(W )) Y − E (Y )( )[ ]+ adE (W − E (W )) Z − E (Z)( )[ ]+bcE (X − E (X )) Y − E (Y )( )[ ]+ bdE (X − E (X )) Z − E (Z)( )[ ] = acCov(W ,Y ) + bcCov(X,Y ) + adCov(W ,Z) +bdCov(X,Z)Summer 201313.7.26 3 Stat 100/Sanchez-Intro Probability 7 € In general, If U = a + bii=1n∑Xi and V = c + djj =1m∑Yj, then Cov(U,V ) = bij =1m∑i=1n∑djCov(Xi,Yj)Summer 2013 4.Multinomial distribution Stat 100/Sanchez-Intro Probability 8 The multinomial distribution is a generalization of the binomial distribution to the situation where we have k > 2 categories. For example, if we are interested in the probability that in a group of 100 people 20 are from California, 30 are from Nevada, 40 are from Texas and 10 are from the rest of the country, and there a probability associated with each category, then we would use the multinomial distribution to answer the question. The assumptions are that a person can fall only in one category, that is, the categories are mutually exclusive. To compute the probabilities, you just use the following joint probability distribution where he probabilities have to sum up to 1 and the x’s have to sum up to n (see next page). Summer 2013 Stat 100/Sanchez-Intro Probability 9 € P(X1= x1, X2= x2,.....Xk= xk) =nx1, x2,...xk⎛ ⎝ ⎜ ⎞ ⎠ ⎟ px1px2......pxk =n!x1!x2!.....xk!px1px2......pxk Summer 201313.7.26 4 Example • With the recent emphasis on solar energy, solar radiation has been carefully monitored at various sites in Florida. Among typical July days in Tampa, 30 percent have total radiation of at most 5 calories, 60 percent have total radiation of at most 6 calories, and 100 percent have total radiation of at most 8 calories. A solar collector for a hot water system is to be run for 6 days. Find the probability that 3 days will produce no more than 5 calories each, 1 day will produce between 5 and 6 calories, and 2 days will produce between 6 and 8 calories. What assumptions must be true for your answer to be correct? Show work. X – number of days having total radiation of at most 5 calories (5 or less), Y = number having total between 5 and 6 W =total between 6 and 8. (X,Y,W) is Multinomial(n=6, px =0.3, py=0.6-0.3=0.3, pw = 1-0.6 =0.4) P(X=3, Y=1, W=2)=[ 6!/(3! 1! 2!)] (0.3)3 (0.3)1 0.42 =0.07776 • Summer 2013 Stat 100/Sanchez-Intro Probability 10 Example • A warehouse contains TV sets, of which 5% are defective, 60% are in working condition but used, and the rest are brand new. What is the probability that in a random sample of five TV sets from this warehouse, there are exactly one defective and exactly two brand new sets? Show work. P(X=1, Y=2, W=2)= Stat 100/Sanchez-Intro Probability 11 Summer 2013 New today • Joint continuous distributions • Marginal distributions • Conditional distributions • Expectations and variances (marginal and conditional) • Probabilities of events defined on a surface. • Covariance and correlation. • More examples on joint probabilities. Stat 100/Sanchez-Intro Probability 12 Summer 201313.7.26 5 Stat 100/Sanchez-Intro Probability 13 1 Joint Probabilities. Suppose we are interested in the joint behavior of two continuous random variables, X and Y, where X and Y might represent the amounts of two different hydrocarbons found in an air sample taken for a pollution study, for example. The relative frequency of these two r.v.’s can be modeled by a bivariate function f(x, y), which forms a probability or relative frequency surface in three dimensions. The probability that X lies in one interval and that Y lies in another interval is then represented as a volume under this surface. € P(a1≤ X ≤ a2,b1≤ Y ≤ b2) = f (x, y)dxdyb1b2∫a1a2∫Summer 2013 Stat 100/Sanchez-Intro Probability 14 Example. A certain process for production of an industrial chemical yields a product that contains two main types of impurities. For a certain volume of sample from this process, let X denote the proportion of total impurities in the sample, and let Y denote the proportion of type I impurity among all impurities found. Suppose that under investigation of many such samples, the join
View Full Document
Unlocking...