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Notes and problems for Econometrics I Question #1: What is a probability density function (pdf)? A pdf on support C is any stable function f(x) which satisfies the following two properties: (1) f ( x ) 0 on C and f ( x ) 0 off C and (2) Question #2: C f ( x )dx 1 What is probability? We can measure probability as P[ X ] f ( x)dx Question #3: What is the expected value of X, which we write as E[X]. Define the expected value of X (or the population mean) as E[ X ] xf ( x)dx . Question #4: What is the variance of X, which we write as var[X]? Define the (population) variance of X as var( X ) ( x E[ X ]) 2 f ( x )dx Question #5: Show that the Question #6: What is var[X] = E[X2] – {E[X]}2. the joint pdf of X and Y (DIY) on support C, which we can write as f ( x, y ) ? The function f(x,y) is a joint pdf if it satisfies two conditions: (1) f ( x, y ) 0 on C f ( x, y ) 0 off and C and (2) C f ( x, y )dxdy 1 Question #7: Suppose we have the random variables X and Y. What is the definition of the covariance of X and Y, which we write as cov[X,Y]? Define the (population) covariance of X and Y as cov( X , Y ) E[{ X E ( X )}{Y E (Y )}] ( x E[ X ])( y E[Y ]) f ( x, y ) dx dy C Note: cov[X,Y] = E[XY] – E[X]E[Y] Exercises: #1. Discuss whether each f(x) is or is not a pdf. a. f(x) = x-1 0 < x 1, α > 0 c. f(x) = exp{-0.5x2} - < x < e. f(x) = ln{x} 0 < x < 1 b. f(x) = 1/2 0 x 1 d. f(x) = exp{-x} 0 x < f. f(x) = (1/ln(2))(1/x) 1 x 2 #2. Find the expected values of X, given the pdf’s below for X. a. f(x) = x-1 0 < x 1, α > 0 c. f(x) = 1 0x1 e. f(x) = 0.5 -1 x 1 b. f(x) = exp{-x} 0 x < d. f(x) = {1/2}0. 5exp{-0.5x2} - < x < f. f(x) = (1/ln(2))(1/x) 1 x 2 #3. Find the variances of W, given the pdf’s below for W a. f(w) = w-1 0w1α>0 b. c. f(w) = {1/2}0. 5exp{-0.5w2} - < w < d. f(w) = exp{-w} 0 w < f(w) = (1/ln(2))(1/w) 1w2 #4. Show that the following functions given below are joint pdf’s for U and V: a. f(u,v) = 1 (u,v) [0,1] X [0,1] b. f(u,v) = (2/ln(2))u/v (u,v) [0,1] X [1,2] c. f(u,v) = {1/2}exp{-0.5(u2 + v2)} (u,v) R2 #5. Show that the covariances of P and Q, given the joint pdf’s below, are zero: a. f(p,q) = {1/2}exp{-0.5(p2 + q2)} (p,q) R2 b. f(p,q) = (2/ln(2))p/q (p,q) [0,1] X [1,2]