Ch 13 Terms and Concepts

what are the standard units of X - µ_X? (Data 8)

(X - µ_X) / SD(X)

What are the 7 properties of Covariance?

1. Constants don't vary: Cov(X,c) = 0 2. Variance is a covariance: Var(X) = Cov(X,X) 3. Covariance is Symmetric: Cov(X,Y) = Cov(Y,X) 4. Covariance is an expectation: it is the expectation of the product of deviations: E[(X - µ_X)(Y - µ_Y)] 5. Independence implies uncorrelated (but covariance of 0 does not imply independence) 6. Addition rule: Cov(X+Y, Z) = Cov(X,Z) + Cov(Y,Z) 7. Bilinearity: Cov(aX, bY) = abCov(X,Y)

What are the two uses of Covariance?

1. Correlation coefficient 2. Finding the variance of a sum: Var(A + B)

Variance(Sn) where Sn = the sum of n random variables (indep or not). 1. What is Var(Sn)? 2. What does Var(Sn) simplify to if X1 through Xn are all independent? 3. What does E(Sn) simplify to in this case? 4. What do Var(Sn) and E(Sn) further simplify to if X1 through Xn are independent AND identically distributed? What about SD(Sn)? 5. For an IID sum of RVs, what are the implications about how E(Sn) and SD(Sn) change as n grows? This has useful applications to sample average, what are they?

1. the sum of all the variances and all the covariances. 2. If these RVs are all independent., then all the covariance terms are 0 and thus, the variance of the sum of n independent RVs is the sum of each of their variances. 3. E(Sn) follows the same simplification as Var(Sn). it is the sum of all individual expectations 4. Var(Sn) = n * Var(X1), E(Sn) = n*E(X1) SD(Sn) = n^1/2 * SD(X1). 5. What this implies is that E(Sn) goes up linearly with n, but SD goes up much more slowly. This means that the expectation of the sample average, E(Xn) where sample average = Xn = 1/n * Sn, is equal to the population average when n increases, and the SD of the sample average, SD(Xn) is equal to the population SD divided by the square root of n. This apparently shows up in the data8 tb and it tells us that as n gets larger, the SD goes to 0.

1. Var(X+Y) = the sum of all the ______ and the sum of all the ________? How many terms are in the first sum? what about the second sum? [13.3.1] 2a. How does the variance of a sum of RVs simplify if there is symmetry in the joint distribution of X1,X2...Xn? (that is, when the RVs are equal in variance and covariance)? [13.4]

1. the sum of all the variances and the sum of all the covariances - The first sum has n terms. - The second sum has n(n−1) terms. 2. Var(Sn) = nVar(X1) + n(n-1)Cov(X1, X2)

Cov(10X - Y, 3Y + Z) =

30Cov(X,Y) + 10Cov(X,Z) +

Let X and Y be two RVs on the same space, and let S = X + Y. What is the deviation of S in terms of the deviations of X and Y? What is expectation of S in terms of expectations of X and Y?

Because D_S = S - E(S) and S = X + Y (given) and E(S) = E(X) + E(Y) by additivity of expectation: (see image) That is, the expectation of S is the sum of the expectation of Y and expectation of X, and the deviation of S is the sum of the deviations of X and Y

Property 4 of covariance allows us to write Cov(X,Y) in terms of the expected product of X and Y, i.e. E(XY). What is that formula? What is this relationship in words?

Cov(X,Y) = E(XY) - µ_Xµ_Y In words, covariance is the mean of the product minus the product of the means (13.2.4).

Cov(X,Y) = E( ____ *____ ) = E[( ____ - _____)(____ - _____)]

E(D_X*D_Y) = E[(X - µ_X)(Y - µ_Y)]

In words, what is the covariance of X and Y?

The expected product of the deviations of X and Y, E(D_X*D_Y)

Relationship between Var(X) and D_x, the deviation from the mean?

The variance of X is the expected deviation from the mean

Ch 13 Ex 1: Var(8X - 9Y + 10)

Var(8X - 9Y + 10) = Var(8X - 9Y) = Var(8X) + Var(9Y) + 2Cov(8X, -9Y) = 64Var(X) + 81Var(Y) + 2(8)(-9)Cov(X,Y) bc indep, cov = 0 Given Var(X) = 2, Var(Y) = 7: = 64Var(X) + 81Var(Y) = 64*2 + 81*7

Given S = X + Y, derive Var(S). In words, what is the variance of S?

Var(S) = Var(X+Y) = E[ (D_X + D_Y)^2 ] foil inside = E[(D_X)^2] + E[(D_Y)^2] + 2E(D_X*D_Y) Because E[(D_X)^2] = Var(X) by definition and same for Var(Y): = Var(X) + Var(Y) + 2E(D_X*D_Y) In words, the variance of S is unlike E(S) and D_S in that the Variance of a sum is in general not the sum of the variances. There's an extra term (the part after the 2 is called the covariance)

Var(X) in terms of expectation?

Var(X) = E[(D_X)^ 2]

What is the covariance of two indicators [13.4.1] When indicator A and indicator B are not independent, what can this tell us?

We can use this thing called the expected product to find the covariance, because the products of indicators are themselves indicators -- indicators of both things occurring, because the product is 1 when BOTH are 1, and is 0 any other time. If they are not independent, then the covariance will not be 0. If its positive, then we can determine if there is positive association of A and B: i.e. given A has occurred, the chance of B is higher

What are the two ways you can calculate Var(2X)?

linear transformation rules of variance from ch 12 applying the formula of the variance of a sum

How can you get rid of the units of covariance? What is this normalized version of Covariance called?

you can standardize the units by dividing Cov(X,Y) by sigma_X*sigma_Y This unit-less version of covariance is correlation