1. 1 G89.2228 Lect 4a G89.2228 Lecture 4a f(X) of special interest: Normal Distribution Are These Random Variables Normally Distributed? Probability Statements and the Normal Distribution Covariance: An important bivariate moment Covariance and. correlation

2. 2 G89.2228 Lect 4a A Density of Special Interest: the Normal Distribution The facts about expectations have been developed without specifying the exact nature of the distribution of X »f(X) can take many different forms »In some cases its form is not known There is one form of f(X) that is of special interest: the normal distribution »The familiar bell shaped distribution so often observed in nature »A distribution that repeatedly emerges in mathematical statistics Central Limit Theorem shows that sums (and averages) of random variables are normally distributed

3. 3 G89.2228 Lect 4a The Normal density A family of distributions that are indexed by two parameters  and  2, the mean and variance  is the index of location, and  2 is the index of spread

4. 4 G89.2228 Lect 4a Normal distributions Why do they appear in nature so often? Linear transformation of X~N( ,  2 ) [X “distributed as” N( ,  2 )] does not change form »If Y=a+bX then Y~N[(  a),(b 2  2 )] »If height is normal in inches, it is normal in centimeters »If self-esteem is normal using one scale, it will usually be normal with a highly correlated scale Empirical operation of Central limit theorem

5. 5 G89.2228 Lect 4a Central Limit Theorem Sums of random variables will be normally distributed as the number of things summed gets large If the distribution of random variables W i is symmetric, “large” may be as little as N=10 »Averages are simply linear transformed sums: (1/n)(  X) Many processes in nature are additive »Height is the sum of annual growths Many psychological measures are additive »Educational achievement as sum of correct test responses

6. 6 G89.2228 Lect 4a Are these random variables normally distributed? Sum five coin flips (H=1, T=0) Sum of fifty coin flips Annual salaries of professors For X~N(  ,  1 2 ) and Y~N(  2,  2 2 ), X+Y For X~N( ,  2 ), X 2 For X~N(  ,  2 ) and Y~N(  2,  2 ), X 2 +Y 2 For X i ~N( ,  2 ) for all i=1,2,...,500,  X i 2 Reaction times to memory trials Errors in smell identification test Sum of 10 attitude strength items

7. 7 G89.2228 Lect 4a Probability statements using the Normal Distribution The distribution of normally distributed random variables, such as sample means, is well known and often presented in tables as N(0,1). Tables can be used by transforming variables with other normal distributions to the form of N(0,1). If X~N(   ) and if  and   are known,then Z = (X-  )/  has N(0,1) distribution This transformation is one-to-one, allowing one to reconstruct X from Z: X =  Z + 

8. 8 G89.2228 Lect 4a Computing Probabilities from N(0,1) Distribution Tables of N(0,1) allow us to ask the probability of sampling Z~N(0,1) in the range (-1, 1). »Pr(-1  Z  1) =.68 If X~N(-.5,.5 2 ) and we want to ask about Pr(-1.5  X  0) we transform to Z and compute

9. 9 G89.2228 Lect 4a One Table Fits All Transformation makes it unnecessary to have all variations of normal curves tabled. The standard normal table describes probability in terms of number of sd's from mean.

10. 10 G89.2228 Lect 4a Assessing Non-independence: One More Expectation Operator Very often we consider two random variables together »height and weight »reaction time and response errors »depression and anxiety »Subject 1 and a yoked control E[(X-  x )(Y-  y )] = Cov(X,Y) =  XY is called the population covariance. Cov(X,Y) measures linear association between the variables It is an expectation that depends on the joint bivariate density of X and Y, f(X,Y). »f(X,Y) says how likely are any pair of values of X and Y

11. 11 G89.2228 Lect 4a Interpreting covariance as a parameter When X and Y tend to increase together, Cov(X,Y)>0 When high levels of X go with low levels of Y, Cov(X,Y)<0 When X and Y are independent, Cov(X,Y) = 0. Note that there are cases when Cov(X,Y) take the value zero when X and Y are related nonlinearly. X Y +,+ -,- -,+ +,-

12. 12 G89.2228 Lect 4a Correlation and Covariance Besides noticing its sign and whether it is zero, it is difficult to interpret the absolute magnitude of covariance Note that Cov(X,Y) is bounded by V(X) and V(Y): If V(X) and V(Y) can be transformed so that both have variances equal to one, then the new covariance is bounded by -1 and +1 »In this case the covariance = correlation,  XY = Corr(X,Y) »It has all the same properties of covariances just discussed, but is easier to interpret

13. 13 G89.2228 Lect 4a Cov (X,Y) as an expectation operator »For k 1 and k 2 as constants, there are facts closely parallel to facts for variances: Cov(k 1 +X, k 2 +Y) = Cov(X,Y) =  XY Cov(k 1 X, k 2 Y) = k 1 *k 2 *Cov(X,Y) = k 1 *k 2 *  XY »Important special case: Let Y * = (1/  Y )Y and X * = (1/  X )X V(X * ) = V(Y * ) = 1.0 Cov(X *,Y * ) = (1/  Y ) (1/  X )  XY =  XY Cov (X *,Y * ) is the population correlation for the variables X and Y,  XY »Since  XY = (1/  Y ) (1/  X )  XY,  XY = (  Y ) (  X )  XY

14. 14 G89.2228 Lect 4a One Payoff for Studying Covariance We can generalize the rule for calculating the variance of a sum of two variables. For any X and Y, Var(X+Y) = V(X) + V(Y) +2Cov(X,Y) Var(X  Y) = V(X) + V(Y)  2Cov(X,Y) More generally, Var(k 1 X+k 2 Y) = k 1 2  X 2 + k 2 2  Y 2 +2k 1 k 2  XY