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Course: MATHCS Math 316, Fall 2010
School: Emory
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361, Math Problem set 5 Due 10/04/10 1. (1.6.8) Let X have the pmf p(x) = ( 1 )x , x = 1, 2, 3, . . . , and zero else2 where. Find the pmf of Y = X 3 . 2. (a) Pick a card from a standard deck. Let X denote the rank of the card(counting ace as one, J=11, Q=12, K=13.) Find the pmf of X . (b) Pick two cards from a deck, with replacement. Let Y denote the highest rank picked. Find the pmf of Y . 3. (1.7.8) A mode of...

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361, Math Problem set 5 Due 10/04/10 1. (1.6.8) Let X have the pmf p(x) = ( 1 )x , x = 1, 2, 3, . . . , and zero else2 where. Find the pmf of Y = X 3 . 2. (a) Pick a card from a standard deck. Let X denote the rank of the card(counting ace as one, J=11, Q=12, K=13.) Find the pmf of X . (b) Pick two cards from a deck, with replacement. Let Y denote the highest rank picked. Find the pmf of Y . 3. (1.7.8) A mode of a distribution of one random variable X is a value of x that maximizes the pdf or pmf. For X of the continuous type, f (x) must be continuous. If there is only one such x, it is called the mode of the distribution. Find the mode of each of the following distributions: p(x) (a) = ( 1 )x , x, 1, 2, 3, . . . , zero elsewhere. 2 (b) f (x) = 12x2 (1 x), 0 < x < 1, zero elsewhere. (c) f (x) = 1 x2 ex , 0 < x < , zero elsewhere. 2 4. (1.7.14) Let X have the pdf f (x) = 2x, 0 < x < 1, zero elsewhere. 3 Compute the probability that X is at least 4 given that X is at least 1 . 2 5. (1.7.17) Divide a line segment into two parts by selecting a point at random. Find teh probability that the larger segment is at least 3 times the shorter. Assume the point is chosen uniformly. 1 6. (1.7.22) Let X have the uniform pdf fX (x) = for < x < . Find 2 2 the pdf of Y = tan(X ). This is the pdf of a Cauchy distribution. 1
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Emory - MATHCS - Math 316
Math 361, Problem set 5Due 10/04/10 1. (1.6.8) Let X have the pmf p(x) = ( 1 )x , x = 1, 2, 3, . . . , and zero else2 where. Find the pmf of Y = X 3 . Answer: Y has pmf p(x) = ( 1 )x/3 for x = 1, 8, 27, . . . , and zero elsewhere. 2 2. (a) Pick a card fr
Emory - MATHCS - Math 316
Math 361, Problem set 6Due 10/18/10 1. (1.8.3) Let X have pdf f (x + 2)/18 for 2 &lt; x &lt; 4, zero elsewhere. Find E[X ], E[(X + 2)3 ] and E[6X 2(X + 2)3 ]. 2. (1.8.5) Let X be a number selected uniformly random from a set of numbers cfw_51, . . . , 100. App
Emory - MATHCS - Math 316
Math 361, Problem set 7Due 10/25/10 1. (1.9.6) Let the random variable X have E[X ] = , E[(X )2 ] = 2 and mgf M (t), h &lt; t &lt; h. Show that E and E exp t X = et/ M t , h &lt; t &lt; h. X = 0, E X 2=1(Recall: exp(x) = ex ). 2. (1.9.7) Show that the moment gene
Emory - MATHCS - Math 316
Emory - MATHCS - Math 316
Emory - MATHCS - Math 316
Emory - MATHCS - Math 316
Midterm Exam IMath 361 9/27/10 Name: Read all of the following information before starting the exam: READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY! Show all work, clearly and in order, if you want to get full credit. I reserve the right to take o point
Emory - MATHCS - Math 316
Midterm Exam IMath 361 9/27/10 Name: Read all of the following information before starting the exam: READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY! Show all work, clearly and in order, if you want to get full credit. I reserve the right to take o point
Emory - MATHCS - Math 316
Midterm Exam IMath 361 9/27/10 Name: Read all of the following information before starting the exam: READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY! Show all work, clearly and in order, if you want to get full credit. I reserve the right to take o point
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 NOTES ON PRICE ELASTICITY OF DEMAND The aim of this note is to go over some of the derivations regarding the price elasticity of demand we covered in class today. Most of it is in Varian The notion of price elasticity is, strictly
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 Drawing Indifference Curves for Min Utility functions Suppose youre given a utility function u(x1, x2) = mincfw_x1, 2x2, i.e., the minimum of the two values x1 or 2x2. Here is how one draws the indifference curves that correspond t
Emory - ECONOMICS - ECON 201
DECOMPOSITIONOFP RICEE FFECTI NTOI NCOMEA NDS UBSTITUTIONE FFECTSx2x2x2C CunAA C BAun un uoB BBn x1 BouoBn x1 BouoBn x1BoA to B : substitution effect (SE) B to C : income effect (IE) A to C : price effect (PE) \$ \$ \$demandpoApo
Emory - ECONOMICS - ECON 201
ECON 201 Shomu Banerjee A NERDY NOTE ON THE RELATIONSHIP BETWEEN AVERAGES AND MARGINALS The aim of this note is to provide the technical details behind the relationship between averages and marginals that we covered in class. This material is once again p
Emory - ECONOMICS - ECON 201
\$ MC_ OACq Total Cost _ OPBq Total Revenue _ APBC Profits_ P AAC B MR = ARSupernormal ProfitsCO \$_ qqMC AC_ OPBq Total Cost _ OPBq Total Revenue_ PBMR = ARNormal ProfitsO_ qq\$ MC_ OACq Total Cost _ OPBq Total Revenue _ APBC Losses ACA
Emory - ECONOMICS - ECON 201
ECON 201 Shomu Banerjee CONSUMPTION EXTERNALITIES &amp; INEFFICIENCY The First Fundamental Welfare Theorem that we studied earlier states that : Under fairly weak assumptions1 about consumer preferences, at a Walrasian equilibrium, the allocation attained is
Emory - ECONOMICS - ECON 201
ECON 201 Shomu Banerjee NOTES ON COASES THEOREM Ronald Coase won the 1991 Economics Nobel Prize &quot;for his discovery and clarification of the significance of transaction costs and property rights for the institutional structure and functioning of the econom
Emory - ECONOMICS - ECON 201
ECON 201 Shomu Banerjee TAX QUIZ The market for a product has the following inverse demand and supply functions Pd = 120 - Qd Ps = 0.5Qs. (a) Find the equilibrium price P* and quantity Q*.(b) Suppose the state government levies a tax of \$15 on each unit
Emory - ECONOMICS - ECON 201
ECON 201 Shomu Banerjee TAX QUIZ ANSWERS The market for a product has the following inverse demand and supply functions Pd = 120 - Qd Ps = 0.5Qs. (a) Find the equilibrium price P* and quantity Q*. Set Pd = Ps and solve for Q: 120 - Q* = 0.5Q* Q* = 80. Sub
Emory - ECONOMICS - ECON 201
ECON 201 Shomu Banerjee ELASTICITY QUIZ (a) The equation for an inverse demand function is given by a Pd = Qd where a &gt; 0 is a constant. Calculate the price elasticity of demand, .!(b) The equation for a linear inverse supply function is given by Ps = a
Emory - ECONOMICS - ECON 201
ECON 201 Shomu Banerjee ELASTICITY QUIZ ANSWERS (a) The equation for an inverse demand function is given by a Pd = Qd where a &gt; 0 is a constant. Calculate the price elasticity of demand, .! dQd (Q )2 dPd a =&quot; d . The slope of the inverse demand is , so =
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 BUDGET QUIZ Milton wishes to spend all his weekly income of \$10 on two goods x and y. Assume both goods are divisible. The price of each good is \$1 per unit. Last Sunday he cut a coupon from the newspaper. It said Buy one unit of y
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 BUDGET QUIZ ANSWER Milton wishes to spend all his weekly income of \$10 on two goods x and y. Assume both goods are divisible. The price of each good is \$1 per unit. Last Sunday he cut a coupon from the newspaper. It said Buy one un
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 BUDGET QUIZ #2 Draw Joan's budget constraint for pies (x) and champagne (y) given the following information: her monthly income is \$100, py = \$5/bottle, px is \$5/pie if she buys between zero and 10 pies and \$4/pie if she buys more
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 BUDGET QUIZ #2 ANSWER Draw Joan's budget constraint for pies (x) and champagne (y) given the following information: her monthly income is \$100, py = \$5/bottle, px is \$5/pie if she buys between zero and 10 pies and \$4/pie if she buy
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201INDIFFERENCE CURVES1. Yoshimitsus utility function over two goods x and y is given to be u(x, y) = x + mincfw_x, y. Draw the indifference curve that gives him 6 units of satisfaction and show the direction in which Yoshimitsus uti
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 INDIFFERENCE CURVES ANSWERS 1. Yoshimitsus utility function over two goods x and y is given to be u(x, y) = x + mincfw_x, y. Draw the indifference curve that gives him 6 units of satisfaction and show the direction in which Yoshimi
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 DEMAND FUNCTION QUIZ #1 Ambrose consumes two goods, x1 and x2, spending all his income m on them. The price of x1 is p1, while that of x2 is p2. His utility function is u(x1, x2) = 4 x1 + x2. The marginal utility for x1 is MU1=2/ x
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 DEMAND FUNCTION #1 ANSWERS Ambrose consumes two goods, x1 and x2, spending all his income m on them. The price of x1 is p1, while that of x2 is p2. His utility function is u(x1, x2) = 4 x1 + x2. The marginal utility for x1 is MU1=2
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 DEMAND FUNCTION QUIZ #2 Anus utility function over two goods x1 and x2 is given to be u(x1, x2) = 2(x1)0.5 + 2(x2)0.5. Her marginal utility for the first good is MU1 = 1/(x1)0.5 while her marginal utility for the second good is MU2
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 DEMAND FUNCTION QUIZ #2 ANSWERS Anus utility function over two goods x1 and x2 is given to be u(x1, x2) = 2(x1)0.5 + 2(x2)0.5. Her marginal utility for the first good is MU1 = 1/(x1)0.5 while her marginal utility for the second goo
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 INCOME, SUBSTITUTION &amp; PRICE EFFECTS PROBLEM 1. Joan's monthly income is \$120, py = \$20/bottle, px is \$12/pie. Her preferences for pies and champagne can be represented by the following utility function: u(x, y) = x + y. (a) Draw J
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 INCOME, SUBSTITUTION &amp; PRICE EFFECTS ANSWERS 1. Joan's monthly income is \$120, py = \$20/bottle, px is \$12/pie. Her preferences for pies and champagne can be represented by the following utility function: u(x, y) = x + y. (a) Draw J
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 INCOME, SUBSTITUTION &amp; PRICE EFFECTS PROBLEM #2 Ysidros typical indifference curve is drawn below. Note that it is piecewise linear with kinks that lie along the two rays from the origin. (a) Ysidros income is \$204, the price of x
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 INCOME, SUBSTITUTION &amp; PRICE EFFECTS #2 ANSWERS Ysidros typical indifference curve is drawn below. Note that it is piecewise linear with kinks that lie along the two rays from the origin. (a) Ysidros income is \$204, the price of x
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201ya xbEXCHANGEObOaxayb Good x is bars of chocolate (assumed to be divisible) and good y is hours spent on computer games. Adam (person 1) has 4 bars of chocolate but doesnt have a computer; Billy (person 2) is allowed by his pa
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201ya xbEXCHANGE ANSWERSObEOaxayb Good x is bars of chocolate (assumed to be divisible) and good y is hours spent on computer games. Adam (person 1) has 4 bars of chocolate but doesnt have a computer; Billy (person 2) is allowed
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201ya xbEXCHANGE #2Ob2010Oa10203040a50xayb In the Edgeworth box above, Andreas origin is O while Beverlys is Ob. Andreas endowment is (10, 20) and Beverlys is (50, 10). They have identical utility functions u( x , y ) = x
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201ya xbEXCHANGE #2 ANSWERSOb2010EOa10203040a50xayb In the Edgeworth box above, Andreas origin is O while Beverlys is Ob. Andreas endowment is (10, 20) and Beverlys is (50, 10). They have identical utility functions u( x
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 EXCHANGE WITH MORE THAN 2 PERSONS An economy consists of two types of consumers, males and females. There are 4 males and 8 females. There are two consumption goods, x1 and x2, that can be traded. There is no production. Each male
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 ANSWERS TO EXCHANGE WITH MORE THAN 2 PERSONS An economy consists of two types of consumers, males and females. There are 4 males and 8 females. There are two consumption goods, x1 and x2, that can be traded. There is no production.
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 RETURNS TO SCALE Find the returns to scale for the following production functions where y denotes output, and K, L, and E are inputs. Show your work! (a) y = K1/3 .L1/3(b) y = 4K.L.E(c) y = [0.3K1/2 + 0.7L1/2]2(d) y = (2K+3L)1/2
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201RETURNS TO SCALE ANSWERSFind the returns to scale for the following production functions where y denotes output, and K, L, and E are inputs. Show your work! (a) y = K1/3 .L1/3 yo = Ko1/3 . Lo1/3 yn = (tKo)1/3 . (tLo)1/3 yn = (t)1/
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 COST MINIMIZATION Suppose a firm has a production function: y = [mincfw_x1, 3x2]1/2 where y is the output and x1 and x2 are inputs. (a) What is the returns-to-scale for this production function?(b) In the graph below, an isoquant
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 COST MINIMIZATION ANSWERS Suppose a firm has a production function: y = [mincfw_x1 , 3x2 ]1/2 where y is the output and x1 and x2 are inputs. (a) What is the returns-to-scale for this production function? yo = [mincfw_x1 , 3x2 ]1/2
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 COST MINIMIZATION #2 Suppose a firm has a production function: y = 2x1 + x2 where y is the output and x1 and x2 are inputs. (a) In the graph below, an isoquant for y = 8 is shown. The prices of the inputs are w1 = \$1 and w2 = \$2. D
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 COST MINIMIZATION #2 ANSWERS Suppose a firm has a production function: y = 2x1 + x2 where y is the output and x1 and x2 are inputs. (a) In the graph below, an isoquant for y = 8 is shown. The prices of the inputs are w1 = \$1 and w2
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 COST MINIMIZATION #3 Suppose a firm has a production function: 1 y = x1 x2 = x1 /2 x1 /2 2 where y is the output and x1 and x2 are inputs. The marginal productivity for the inputs is given by x1 /2 x1 /2 MP = 21 /2 and MP2 = 11 /2
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 COST MINIMIZATION #3 ANSWERS Suppose a firm has a production function: 1 y = x1 x2 = x1 /2 x1 /2 2 where y is the output and x1 and x2 are inputs. The marginal productivity for the inputs is given by x1 /2 x1 /2 MP = 21 /2 and MP2
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201PERFECT COMPETITION QUIZ1 A perfectly competitive firm has a cost function given by C(q) = 72 + 2q + 2 q2. The marginal cost for this firm is MC(q) = 2 + q. The market price, P, for its product is \$20. (a) Find out the quantity th
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201PERFECT COMPETITION ANSWERS1 A perfectly competitive firm has a cost function given by C(q) = 72 + 2q + 2 q 2. The market price, P, for its product is \$20. (a) Find out quantity the firm produces in the short-run, q*. Set P = MC t
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 PERFECT COMPETITION CONTD QUIZ Recall the perfectly competitive firm with a cost function given by C(q) = 72 + 2q + 0.5q2. In the previous quiz, you had calculated the firms supply curve to be qs(P) = P 2 for P &gt; 2 and zero otherwi
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 PERFECT COMPETITION CONTD ANSWERS Recall the perfectly competitive firm with a cost function given by C(q) = 72 + 2q + 0.5q2. In the previous quiz, you had calculated the firms supply curve to be qs(P) = P 2 for P &gt; 0 and zero othe
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 MONOPOLY Trygve is a monopolist with a cost function given by 1 C(Q) = 72 + 100Q + 2 Q2 facing an inverse market demand curve P = 700 - Q. Its marginal cost is MC = 100 + Q. (a) Find his profit-maximizing price and output combinati
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 MONOPOLY ANSWERS Trygve is a monopolist with a cost function given by 1 C(Q) = 72 + 100Q + 2 Q2 facing an inverse market demand curve P = 700 - Q. Its marginal cost is MC = 100 + Q. (a) Find his profit-maximizing price and output c
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 MONOPOLY WITH TAX QUIZ A monopolist producer faces the (inverse) demand curve given by P = a - Q, where P is the price charged and Q is the quantity demanded. The total cost of producing the good is C(Q) = cQ; the marginal cost is
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 MONOPOLY WITH TAX QUIZ ANSWERS A monopolist producer faces the (inverse) demand curve given by P = a - Q, where P is the price charged and Q is the quantity demanded. The total cost of producing the good is C(Q) = cQ; the marginal
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 CONSUMPTION EXTERNALITY Suppose Atkins and Bodkins are the only two consumers in our economy, there are two goods x and y, and Atkins endowment is a = (10, 0) while Bodkins is b = (0, 10). Their utility functions are given by uA(xa
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 CONSUMPTION EXTERNALITY ANSWERS Suppose Atkins and Bodkins are the only two consumers in our economy, there are two goods x and y, and Atkins endowment is a = (10, 0) while Bodkins is b = (0, 10). Their utility functions are given
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 EXTERNALITY QUIZ A honey farm is located next to an apple orchard. The honey producers cost function is cH(h) = (h2/100) a/5, where h is the quantity of honey produced and a is the quantity of apples producedthere is a positive ext
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201 EXTERNALITY QUIZ ANSWERS A honey farm is located next to an apple orchard. The honey producers cost function is cH(h) = (h2/100) a/5, where h is the quantity of honey produced and a is the quantity of apples producedthere is a posi
Emory - ECONOMICS - ECON 201
Shomu Banerjee ECON 201GROUP PRICING QUIZA monopolist engages in third degree differential pricing (i.e., group pricing) across two markets with demand curves given by and Its cost function is given by P1 = 60 0.5Q1 P2 = 80 Q2. C(Q) = Q2,where Q = Q1 +