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intermediate microeconomics

Microeconomic Theory Basic Principles And Extensions 1st Edition Christopher M Snyder, Walter Nicholson, Robert B Stewart - Solutions

11.1 Suppose there are 50 identical firms in a perfectly com petitive industry. Each firm has a short-run total cost function of the form TC = 0.2q 2 + 5q + 10.a.Calculate the irm’s short-run supply curve with q as a function of market price (P).b.c.On the assumption that there are no interaction
10.8 Some envelope results Young’s theorem can be used in combination with the envelope results to derive some useful results.a.Show that ∂l (P, v, w) /∂v = ∂k (P, v, w) /∂w using substitution and output effects.b.c.d.Use the result from part (a) to show how a unit tax on labour would be
10.7 How would you expect an increase in output price, P, to affect the demand for capital and labour inputs?a.Explain graphically why, if neither input is inferior, it seems clear that a rise in P must not reduce the demand for either factor.b.c.Show that the graphical presumption from part (a)is
10.6 10.9 Le Châtelier’s Principle This problem concerns the relationship between demand and marginal revenue curves for a few functional forms.a.Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any
10.5 Supposing g γ = 0.5, calculate the irm’s total cost function and proit function.If v = 1000, w = 500 and P = 600, how many students will QuickLearn train and what are its proits?If the price candidate drivers are willing to pay rises to P = 900, how much will proits change?Graph
10.4 What is the proit function for this irm?What is the supply function for assembled calculators [q(P, w)]?What is this irm’s demand for labour function[l(P, w)]?Describe intuitively why these functions have the form they do.QuickLearn Driving School trains learner drivers.The number of
10.3 The production function for a firm in the business of calcu lator assembly is given by q = 2!l, where q denotes finished calculator output and l denotes hours of labour input. The firm is a price-taker both for respect to any input price yields (the negative of) the demand function for that
10.2 Determine Harry’s maximum proit on cement.Graph these results and label Harry’s supply curve.Universal Widget produces high-quality widgets at its plant in Johannesburg, South Africa for sale throughout the world.The cost function for total widget production (q) is given by TC = 0.1q 2 +
10.1 Harry’s Hardware is a small business that sells bags of cement in a market where it is a price-taker and P = MR The prevailing market price of cement is €10 per 50 kg bag. Harry’s total cost function is given by TC = 0.05q 2 + 5q + 22 where q is the number of bags of cement Harry chooses
9.12 Given q, w and v, how should the capital stock be chosen to minimise total cost?Use your results from part (f ) to calculate the long-run total cost of polo stick production.For w = £4, v = £1, graph the long-run total cost curve for polo stick production. Show that this is an envelope for
9.11 Calculate the irm’s long-run total, average and marginal cost functions.Suppose that k is ixed at 10 in the short run.Calculate the irm’s short-run total, average, and marginal cost functions.Suppose v = 1 and w = 3. Calculate this irm’s long-run and short-run average and marginal cost
9.10 If the two outputs are actually the same good, we can deine total output as q = q1 + q2. Suppose that in this case average cost (= C/q) decreases as q increases. Show that this irm also enjoys economies of scope under the deinition provided here.Suppose that a firm’s fixed proportion
9.9 Suppose that a firm produces two different outputs, the quantities of which are represented by q1 and q2. In gen eral, the firm’s total costs can be represented by C(q1, q2).This function exhibits economies of scope if C(q1, 0) +C(0, q2) > C(q1, q2) for all output levels of either
9.8 0 ≤ βi ≤ 1, i = 0,…, 3.If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters β0, … , β3?Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal
9.7 Show that MPk = (q/k)1–ρ and MPl = (q/l)1–ρ.Show that RTS = (k /l)1–ρ; use this to show thatσ = 1/(1 – ρ).Determine the output elasticities for k and l; and show that their sum equals 1.Prove that ql = a�q�l bσand hence that ln aq l b = σ ln a�q�l b .Note: The latter
9.6 Suppose we are given the constant returns-to-scale CES production function q = [k ρ + l ρ]1/ρ.a.b.c.d.
9.5 If a single irm produces bottles in both locations, then it will obviously want to get as large an output as possible for a given labour input. How should the irm allocate labour between the two locations to do so? Explain precisely the relationship between l1 and l2.Assuming that the irm
9.4 Suppose that the quantity of plastic bottles produced (q)takes place in two locations. Capital inputs cannot change and changes in production use only labour as an input. The production function in location 1 is given by q1 = 10l 0.5 1and in the other location by q2 = 50l 0.5 2 .a.b.
9.3 Again if k = 10, graph the MPl curve. At what level of labour input does MPl = 0?Suppose capital inputs double. How would your answers to parts (a) and (b) change?Does the production function for product x exhibit constant, increasing, or decreasing returns to scale?Basil Fawlty is planning to
9.2 Suppose the production function for product x is given by q = kl − 0.8k 2 − 0.2l 2,● All cost curves are drawn on the assumption that the input prices are held constant. When input prices change, cost curves will shift to new positions. The extent of the shifts will be determined by the
9.1 TransFast Trucking Company uses two sizes of trailers for transportation contracts. Smaller trailers can carry 2 tons and larger trailers can carry 4 tons.Output per hour(kg carted)Small trailer Large trailer a.2000 Capital input 14000 2Labour input 11 TransFast Trucking Company is awarded a
8.12 refinements of perfect Bayesian equilibrium Recall the job-market signalling game in Example 8.9.a.Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the irm offers an uneducated worker a job. Be sure to
8.11 Alternatives to Grim strategy Suppose that the Prisoners’ Dilemma stage game (see Figure 8.1) is repeated for infinitely many periods.a.Can players support the cooperative outcome by using tit-for-tat strategies, punishing deviation in a past period by reverting to the stage-game Nash
8.10 rotten Kid theorem In A Treatise on the Family (Cambridge, MA: Harvard University Press, 1981), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1) and the child’s parent (player 2). The child moves first,
8.9 fairness in the Ultimatum Game Consider a simple version of the Ultimatum Game discussed in the text. The first mover proposes a division of €1. Let r be the share received by the other player in this proposal (so the first mover keeps 1 – r), where 0 ≤ r ≤ 1/2. Then the other player
8.8 Indicate the Bayesian–Nash equilibrium on a best-response function diagram.Which type of player 1 would like to send a truthful signal to player 2 if it could? Which type would like to hide his or her private information?In Blind Poker, player 2 draws a card from a standard deck and places it
8.7 Verify that the Nash equilibrium is the usual one for the Prisoners’ Dilemma and that both players have dominant strategies.Suppose the stage game is repeated ininitely many times. Compute the discount factor required for their suspects to be able to cooperate on silent each period. Outline
8.6 Solve for the symmetric mixed-strategy equilibrium. That is, letting p be the probability that a male approaches the blond, ind p*.Show that the more males there are, the less likely it is in the equilibrium from part (b) that the blond is approached by at least one of them. Note: This
8.5 The Academy Award–winning movie A Beautiful Mind about the life of John Nash dramatises Nash’s scholarly contribution in a single scene: his equilibrium concept dawns on him while in a bar bantering with his fellow male graduate students. They notice several women, one blond and the rest
8.4 Two security guards employed by different companies, i = 1, 2, simultaneously choose how many hours li to spend patrolling a street. The average benefit per hour can be expressed as 10−li+lj 2and the (opportunity) cost per hour for each is 4.Security guard i’s average benefit is increasing
8.3 The game of Chicken is played by two macho teens who speed toward each other on a single-lane road. The first to veer off is branded the chicken, whereas the one who does not veer gains peer-group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the Chicken game
8.2 The mixed-strategy Nash equilibrium in the Battle of the Sexes in Figure 8.3 may depend on the numerical values for the payoffs. To generalise this solution, assume that the payoff matrix for the game is given by where K ≥ 1. Show how the mixed-strategy Nash equilibrium depends on the value
8.1 Consider the following game:a. Find the pure-strategy Nash equilibria (if any).b. Find the mixed-strategy Nash equilibrium in which each player randomises over just the irst two actions.c. Compute players’ expected payoffs in the equilibria found in parts (a) and (b).d. Draw the extensive
7.14 the portfolio problem with a normally distributed risky asset In Example 7.3 we showed that a person with a CARA utility function who faces a Normally distributed risk will have expected utility of the form E[U(W)] = μW − (A/2)σ2 W′where μW is the expected value of wealth and σ2 W is
7.13 Graphing risky investments Investment in risky assets can be examined in the state-preference framework by assuming that W * Euros invested in an asset with a certain return r will yield W *(1 + r) in both states of the world, whereas investment in a risky asset will yield W *(1 + rg) in good
7.12 More on the CrrA function For the CRRA utility function (Equation 7.42), we showed that the degree of risk aversion is measured by 1 – R. In Chapter 3 we showed that the elasticity of substitution for the same function is given by 1/(1 – R). Hence the measures are reciprocals of each
7.11 Prospect theory Two pioneers of the field of behavioural economics, Daniel Kahneman and Amos Tversky (winners of the Nobel Prize in economics in 2002), conducted an experiment in which they presented different groups of subjects with one of the following two scenarios:● Scenario 1: In
7.10 HArA Utility The CARA and CRRA utility functions are both members of a more general class of utility func tions called harmonic absolute risk aversion (HARA)functions. The general form for this function is U(W) = θ(μ + W/γ)1−γ, where the various parameters obey the following
7.9 Now consider varying the gamble in part (a) by multiplying each prize by a positive constant k.Let h = kν. What is the value of E(h 2)?Suppose this person has a logarithmic utility function U(W) = ln W. What is a general expression for r (W )?Compute the risk premium (p) for k = 0.5, 1 and 2,
7.8 What mix of wheat and maize would provide maximum expected utility to this farmer?Would wheat crop insurance – which is available to farmers who grow only wheat and which costs€4000 and pays off €8000 in the event of a rainy growing season – cause this farmer to change what he plants?In
7.7 A farmer believes there is a 50–50 chance that the next growing season will be abnormally rainy. His expected utility function has the form expected utility = 1 2 ln YNR + 1 2 ln YR, where YNR and YR represent the farmer’s income in the states of ‘normal rain’ and ‘rainy’,
7.6 If there is a 25 per cent probability that Amy will lose €1000 of her cash on the safari, what is the safari’s expected utility?Suppose that Amy can buy insurance against losing the €1000 (say, by purchasing traveller’s cheques) at an ‘actuarially fair’ premium of€250. Show that
7.5 A three-month safari to the Okavango Delta in Botswana is estimated to cost Amy €10 000. The utility from the safari is a function of how much she actually spends on it (Y), given by U(Y) = ln Y.a.b.c.
7.4 Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable?Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly?A health insurance company considers the risk to a Sherpa
7.3 An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 per cent chance that all the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take
7.2 Show that if an individual’s utility-of-wealth function is con vex then he or she will prefer fair gambles to income cer tainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behaviour is common? What factors might tend to limit its
7.1 George is seen to place an even-money £100 000 bet on Manchester United to win the English Premier League Final. If George has a logarithmic utility-of-wealth func tion and if his current wealth is £1 000 000, what must he believe is the minimum probability that the Manchester United will win?
6.8 separable utility A utility function is called separable if it can be written as U(x, y) = U1(x) + U2(y), where U′ i > 0, Ui″< 0, and U1, U2 need not be the same function.a.What does reparability assume about the cross partial derivative Uxy? Give an intuitive discussion of what word this
6.7 Consumer surplus with many goods The welfare costs of changes in a single price can be measured using expenditure functions and compensated demand curves. This problem asks you to generalise this to price changes in two (or many) goods.a.Suppose that an individual consumes n goods and that the
6.6 Example 6.3 computes the demand functions implied by the three-good CES utility function U(x, y, z) = −1 x − 1 y − 1 z .a.b.Use the demand function for x in Equation 6.32 to determine whether x and y or x and z are gross substitutes or gross complements.How would you determine whether x
6.5 How might one deine a composite commodity for the pensioner’s solid food consumption?State the pensioner’s optimisation problem as one of choosing between solid (s) and liquid (l)food consumption.What is the pensioner’s demand functions for s and l ?Once the pensioner decides how much to
6.4 Explain why ∂c/∂pbt = 0.Is it also true here that ∂c/∂pb and ∂c/∂pt are equal to 0?A pensioner allocates her budget between three commod ities, bread (b), tuna (t) and prune juice (p). If the ratio of the price of tuna to bread is a constant pt pb:a.b.c.d.
● An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utility-providing attributes. This may provide additional insights into relationships among goods.The migrant worker chooses to allocate
6.3 A migrant farm worker consumes only two goods, coffee(c) and buttered toast (bt) that he obtains from the local take-away. Buttered toast is a composite commodity con sisting of a single slice of toast and two grams of butter.● Focusing only on the substitution effects from price changes
6.2 Prove that an increase in the price of mampoer will not inluence JJ’s consumption of steak.Show also that ∂m/∂ps = 0.Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the derivatives in parts (a)and (b) are identical.Prove
6.1 Johannes Jacobus (JJ) receives utility from two goods, mampoer (liquor) (m) and steak (s)U(m, s) = m · s.a.b.c.d.
5.13 Price indifference curves Price indifference curves are iso-utility curves with the prices of two goods on the X- and Y-axes, respectively.Thus, they have the following general form:(p1, p2)∣v(p1, p2, I) = v0.a.Derive the formula for the price indifference curves for the Cobb–Douglas case
5.12 the almost ideal demand system The general form for the expenditure function of the almost ideal demand system (AIDS) is given by nIn E(p1,…, pn, U) = a0 + a i=1 k+ Uβ0q i=1 nαi In pi + 1 2a ni=1 aj=1 pkβk,γij In pi In pj For analytical ease, assume that the following restrictions
5.11 Quasi-linear utility (revisited)Consider a simple quasi-linear utility function of the form U(x, y) = x + In y.a.Calculate the income effect for each good. Also calculate the income elasticity of demand for each good.b.c.d.Calculate the substitution effect for each good.Also calculate the
5.10 Aggregation of elasticities for many goods The three aggregation relationships presented in this chapter can be generalised to any number of goods.This problem asks you to do so. We assume that there are n goods and that the share of income devoted to good i is denoted by si. We also define
5.9 More on elasticities Part (e) of Problem 5.9 has a number of useful applications because it shows how price responses depend ultimately on the underlying parameters of the utility function.Specifically, use that result together with the Slutsky equation in elasticity terms to show:a.b.c.In the
5.8 share elasticities Empirical work in demand theory focuses on income shares.For any good, x, the income share is defined as sx = pxx/I.In this problem we show that most demand elasticities can be derived from corresponding share elasticities.a.Show that the elasticity of a good’s budget share
5.7 Show that the share of income spent on a good x is sx = d In Ed In px, where E is total expenditure.
5.6 Suppose that a person regards ham and cheese as pure complements – he or she will always use one slice of ham in combination with one slice of cheese to make a ham and cheese sandwich. Suppose also that ham and cheese are the only goods that this person buys and that bread is free.a.If the
5.5 Suppose the utility function for goods x and y is given by utility = U(x, y) = xy + y.a.Calculate the uncompensated (Marshallian)demand functions for x and y, and describe how the demand curves for x and y are shifted by changes in I or the price of the other good.b.c.Calculate the expenditure
5.4 Prove that if an individual’s tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions; that is, prove that Giffen’s paradox cannot occur.As in Example 5.1, assume that utility is given by utility = U(x, y) = x 0.3y 0.7.a.Use the
5.3 As defined in Chapter 3, a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: the MRS depends on the ratio y/x.a.Prove that, in this case, ∂x/∂I is constant.b.
5.2 A labourer earns €53 per week and chooses to consume only ham and cheese sandwiches. Ham costs €0.75 a slice and cheese costs €0.50 per slice, bread is provided free of charge by his employer. The labourer spends his entire earnings on ham and cheese sandwiches that comprise 1 slice of
5.1 Bottled water can only be purchased in two different-sized containers: 0.75 litres and 2 litres. Assume the water itself is identical, and these two ‘goods’ are perfect substitutes.a.If the containers themselves yield no utility, express the utility function in terms of quantities of
4.11 Ces indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expendi ture functions. Suppose utility is given by U(x, y) = (xδ + yδ)1/δ[in this function the elasticity of substitution σ = 1/(1
4.10 stone–Geary utility Suppose individuals require a certain level of food (x) to remain alive. Let this amount be given by x0. Once x0 is purchased, individuals obtain utility from food and other goods (y) of the form U(x, y) = (x − x0)αyβ, where α + β = 1.a.Show that if I > px x0 then
4.9 Suppose that we have a utility function involving two goods that is linear of the form U(x, y) = ax + by.Calculate the expenditure function for this utility function. Hint: The expenditure function will have kinks at various price ratios.
4.8 Two of the simplest utility functions are:1.Fixed proportions: U(x, y) = min[x, y]2.Perfect substitutes: U(x, y) = x + y a.For each of these utility functions, compute the following:• Demand functions for x and y• Indirect utility function• Expenditure function Discuss the particular
Use Equation 4.52 to estimate the degree to which good x must be subsidised to increase this person’s utility from U = 2 to U = 3. How much would this subsidy cost the government? How would this cost compare with the cost calculated in part (b)?
Use a graph to show that an income grant to a person provides more utility than does a subsidy on good x that costs the same amount to the government.Use the Cobb–Douglas expenditure function presented in Equation 4.52 to calculate the extra purchasing power needed to increase this person’s
4.6 Suppose that a consumer’s utility function for goods x, y and z is expressed according to the Cobb–Douglas utility function U(x, y, z) = x 0.5y 0.5(1 + z)0.5.Prices are given by px = 1, py = 4, and pz = 8.Consumer income is given by I = 8.a.Determine the utility maximising quantity of good
4.5 Andrew’s utility function for lunch is given by:U(l) = l Andrew prefers his lunch in an exact proportion of two parts gravy (g) to maize meal (m). Hence we can rewrite Andrew’s utility function as U(l) = U(g, m) = minag 2 , mb a.b.c.d.Graph Andrew’s indifference curve in terms of g and m
On a given day, John enjoys the consumption of croissants (c) and butter (b) according to the function U(c,b) = 20c − c2 + 18b − 3b2.How many croissants and portions of butter does he consume during a day? (Assuming John does not face any budgetary constraints.)John has been advised to restrict
When she arrived at the store, our wine consumer discovered that the price of the French Bordeaux had fallen to €30 a bottle. If the price of the Italian Campania wine remains at €15 per bottle, how of consumption of some goods is zero. In this case, the ratio of marginal utility to price for
If the cafe tries to discourage the consumption of toasted cheese sandwiches by increasing the price by £0.50, by how much will Patrick’s parents have to increase his allowance to provide him with the same level of utility he received in part (a)?A wine consumer has €600 to spend to build a
4.2 a.b.If a toasted cheese sandwich costs £1.00 and a soft drink costs £0.25 per cup, how should Patrick spend the £5.00 his parents give him to maximise his utility?
4.1 Each day Patrick, who is in year four, buys lunch at the local cafe. He likes only toasted cheese sandwiches (t)and soft drinks (s), and these provide him a utility of utility = U(t, s) = "ts a.b.
3.15 the benefit function c.In a 1992 article David G. Luenberger introduced what he termed the benefit function as a way of incorporating some degree of cardinal measurement into utility theory.11 The author asks us to specify a certain elementary consump tion bundle and then measure how many
3.14 Preference relations The formal study of preferences uses a general vector notation. A bundle of n commodities is denoted by the vector x = (x1, x2, … , xn), and a preference relation (≻) is defined over all potential bundles. The statement x1 ≻ x2 means that bundle x1 is preferred to
3.13 the quasi-linear function Consider the function U(x, y) = x + ln y. This is a function that is used relatively frequently in economic modelling as it has some useful properties.a.Find the MRS of the function. Now, interpret the result.b.c.d.e.Conirm that the function is quasi-concave.Find the
3.12 Ces utility a.Show that the CES functionαxδδ +βyδδis homothetic. How does the MRS depend on the ratio y/x?b.c.d.e.Show that your results from part (a) agree with our discussion of the cases δ = 1 (perfect substitutes) and δ = 0 (Cobb–Douglas).Show that the MRS is strictly diminishing
3.11 independent marginal utilities Two goods have independent marginal utilities if�2U�y�x = �2U�x�y = 0.Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing MRS. Provide an example to
3.10 Cobb–Douglas utility Example 3.3 shows that the MRS for the Cobb–Douglas function U(x, y) = xαyβ2/6/1 is given by a.MRS = αβ ay x b.Does this result depend on whether α + β = 1?Does this sum have any relevance to the theory of choice?b.c.For commodity bundles for which y = x, how
3.9 initial endowments Suppose that a person has initial amounts of the two goods that provide utility to him or her. These initial amounts are given by x and y.a.Graph these initial amounts on this person’s indifference curve map.b.c.If this person can trade x for y (or vice versa) with other
Find utility functions given each of the following indiffer ence curves [defined by U(·) = k].a.b.c.z = k 1/δxα/δyβ/δ.y = 0.5"x 2 − 4(x 2 − k) − 0.5x.z = "y 4 −4x(x 2y − k)2x− y 22x .
She is indifferent between bundle (6, 5) and bundle(12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve?A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1).She is also willing to trade in 1 unit of x
Many advertising slogans seem to be asserting something about people’s preferences. How would you capture the following fictitious slogans with a mathematical utility function?a.Promise margarine is just as good as butter.b.c.d.e.3.7 a.Things go better with Coke.You can’t eat just one Kettle
3.6 Express FF’s utility function in terms of a single good? What is that good?Suppose foot-long hot dogs cost £2.50 each, rolls cost £0.50 each, mustard costs £0.15 per gram, and cheese costs £0.50 per gram. How much does the good deined in part (b) cost?If the price of foot-long hot dogs
3.5 U(x, y) = min(x, y).U(x, y) = max(x, y).U(x, y) = x + y.The Footie Fanatic (FF) always eats his stadium hotdog in a special way; he uses a foot-long hot dog (h) together with precisely half a roll (r), 1 gram of mustard (m), and 2 grams of cheese (c). His utility is a function only of these
3.4 As we saw in Figure 3.5, one way to show convexity of indifference curves is to show that, for any two points (x1, y1)and (x2, y2) on an indifference curve that promises U = k,the utility associated with the point ax1 + x2 CHAPTER 3 PREFERENCES AND UTILITY 91 2 , y1+y2 2 b is at least as great
3.3 Consider the following utility functions:a.U(x, y) = xy.b.c.U(x, y) = x 2y 2.U(x, y) = ln x + ln y.Show that each of these has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?
In footnote 7 we showed that for a utility function for two goods to have a strictly diminishing MRS (i.e., to be strictly quasi-concave), the following condition must hold:UxxU 2x − 2UxyUxUy + UyyU 2y < 0 Use this condition to check the convexity of the indiffer ence curves for each of the
3.1 Graph a typical indifference curve for the following utility functions, and determine whether they have convex indiffer ence curves (i.e., whether the MRS declines as x increases).a.U(x, y) = 3x + y.b.c.d.e.3.2 U(x, y) = !x · y.U(x, y) = !x + y.U(x, y) = "x 2 − y 2.U(x, y) = xy x + y .
2.16 More on covariances Here are a few useful relationships related to the covari ance of two random variables, xl and x2.a.Show that Cov(x1, x2) = E(x1x2) – E(x1)E(x2). An important implication of this is that if Cov(x1, x2) = 0, E(x1x2) = E(x1)E(x2). That is, the expected value of a product of
2.15 More on variances The definition of the variance of a random variable can be used to show a number of additional results.a.Show that Var(x) = E(x2) – [E(x)]2.b.c.Use Markov’s inequality (Problem 2.14d) to show that if x can take on only non-negative values, P[(x − μx) ≥ k] ≤ σ2 xk

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