Econ 3120 1st Edition Lecture 2 Outline of Last Lecture I Random Variables II Continuous Distributions III Multivariate Distributions Outline of Current Lecture IV Marginal Distributions V Expectations and Variance VI Covariance Current Lecture 1 4 Marginal Distributions Suppose we have a joint distribution of X and Y given by f x y How do we find the distribution of Y alone The marginal distribution of a discrete random variable X is given by g x y f x y Similarly the marginal distribution of Y is given by h y x f x y In the continuous case the marginal distributions for X and Y are given by g x f x y dy and h y f x y dx 1 5 Conditional Distributions The conditional distribution of X is defined as f x y f x y h y when h y is the value of the marginal distribution of Y at y The conditional distribution of Y is defined similarly Given the marginal distribution for X of g x the conditional distribution is given by w y x f x y g x 1 6 Independence Random variables X and Y are independent if and only if f x y f x f y f x y g x Expectation and Variance 2 1 Expectation Definition If X is a discrete random variable and f x represents its probability distrubtion function the expected value or mean of X is given by E X x x x f x For a continuous random variable the expected value is given by E X x x f x dx We can think about the expected value as the weighted average of X The value of each possible realization of X is weighted by the probability that x occurs We can also take expectations of functions of a random variable E w X x w x f x dx Or we can take expectations of functions of multiple random variables E w X Y x w x y f x y dxdy 2 1 1 Properties of Expectations 1 For any constant c E c c 2 For any constants a and b E aX b aE X b 3 If a1 a2 an are constants and X1 X2 Xn are random variables then E n i 1 aiXi aiE Xi These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute 2 2 Variance While the expectation tells us about the average of X the variance gives us information on the dispersion of X around the mean The variance is defined as Var X 2 x E X E X 2 It turns out that this is equivalent to Var X E X 2 E X 2 The variance is the average squared deviation from the mean We often think about dispersion in terms of standard deviation which is defined as sd X x p Var X 2 2 1 Properties of the variance 1 Var X 0 2 Var X 0 if and only if X can take on a single value 3 For any constants a and b Var aX b a 2Var X 2 3 Covariance When we are dealing with more than one random variable it is useful to summarize how these two random variables move together Suppose we have two random variables X and Y and we define E X x and E Y y The covariance between these two random variables is defined as Cov X Y xy E X x Y y We can also write the covariance as Cov X Y E X x Y E X Y y E XY x y 2 3 1 Properties of the covariance 1 If X and Y are independent then Cov X Y 0 This follows since E XY E X E Y when X and Y are independent 2 For any constants a1 b1 a2 b2 Cov a1X b1 a2Y b2 a1a2Cov X Y Now that we know about the Covariance we can define a third property of the variance 3 Var aX bY a 2Var X b 2Var Y 2abCov X Y One issue with covariance is that the units are difficult to interpret It turns out that we can scale covariance by the standard deviations of both variables and to get the unit friendly correlation Corr X Y xy Cov X Y sd X sd Y xy x y