10. Hypothesis Testing Methods and Confidence Regions 609
3.12 Appendix: Proofs and Conditions for Positive Semidefiniteness
3.12.1 Proof of Theorem 3.2
Discrete Case LetYẳg(X). The density function for the random variableYcan be represented by:
hðyị ẳPYðyị ẳPXðfx:gðxị ẳy;x2RðXịgị ẳ X
x:gðxịẳy
f g
fðxị:
That is, the probability of the outcome y is equal to the probability of the equivalent event {x:g(x)ẳy}, which is the inverse image ofy. Then
EðgðXịị ẳEðYị ẳ X
y2RðYị
yhðyị ẳ X
y2RðYị
y X
x:gðxịẳy
f g
fðxị
ẳ X
y2RðYị
X
x:gðxịẳy
f g
gðxịfðxị ẳ X
x2RðXị
gðxịfðxị;
where the next to last expression is true, since g(x)ẳy for allx∈{x :g(x)ẳy}, and the last expression is true since P
y2RðYị
P
x:gðxịẳy
f ggðxịfðxịis equivalent to 162 Chapter 3 Expectations and Moments of Random Variables
summing over allx∈R(X) because the collection of ally∈R(Y) (the outer sum) is
the setR(Y)ẳ{y:yẳg(x),x∈R(X)}. n
Continuous Case To prove the theorem for the continuous case, we first need to establish the following lemma.
Lemma 3.1 For any continuous random variableY, the expectation ofY, if it exists, can be written as
EðYị ẳ Z 1
0 Pðy>zịdz Z 1
0 Pðy zịdz:
Proof of Lemma Leth(y) be the density function ofY. ThenP(y>z)ẳR1
z hðyịdy, so that Z 1
0
Pðy>zịdzẳ Z 1
0
Z 1
z
hðyịdydzẳ Z 1
0
Z y 0
dz
hðyịdyẳ Z 1
0
yhðyịdy; where the second equality was simply the result of changing the order of integration (note that the inner range of integration is a function of the outer range of integration, and thesameset of (y,z) points are being integrated over).
Similarly,P(y z)ẳ Rz
1h(y)dy, so that Z 1
0
Pðy zịdzẳ Z 1
0
Z z
1hðyịdydzẳ Z 0
1
Z y 0
dz
hðyịdyẳ Z 0
1yhðyịdy:
Therefore, Z 1
0
Pðy>zịdz Z 1
0
Pðy zịdzẳ Z 1
0
yhðyịdyỵ Z 0
1yhðyịdyẳEðYị: n Note that the lemma (integrals and all) also applies to discrete random variables.19
Using the lemma, we have EðgðXịị ẳ
Z 1
0
P gð ðxị>zịdz Z 1
0
P gð ðxị zịdz
ẳ Z 1
0
Z
fx:gðxị>zgfðxịdxdz Z 1
0
Z
fx:gðxị zgfðxịdxdz
ẳ Z
fx:gðxị>0g
Z gðxị 0
dz
" #
fðxịdx Z
fx:gðxị 0g
Z gðxị 0
dz
" # fðxịdx
ẳ Z
fx:gðxị>0ggðxịfðxịdxỵ Z
fx:gðxị 0ggðxịfðxịdx:
ẳ Z 1
1gðxịfðxịdx: n
19See P. Billingsley (1986)Probability and Measure, 2nd ed. New York: John Wiley, pp. 73–74 for the method of proof in the discrete case.
3.12.2 Proof of Theorem 3.4 (Jensen’s Inequalities)
We prove the result for the convex case. The proof of the concave case is analogous, with inequalities reversed.
Ifgis a convex function forx∈I, then there exists a line going through the point E(X), say‘ðxị ẳaỵbx, such thatgðxị ‘ðxị ẳaỵbx 8x2Iandg(E(X)) ẳ a +bE(X) (see Figure3.6). Now note that
EðgðXịị ẳ X
x2RðXị
gðxịfðxị X
x2RðXị
aþbx ð ịfðxị
ẳaỵbEðXị ẳgðEðXịị (discrete) EðgðXịị ẳ
Z 1
1gðxịfðxịdx Z 1
1ðaỵbxịfðxịdx
ẳaỵbEðXị ẳgðEðXịị (continuous) sinceg(x) a+ bx8x∈I,20so that E(g(X))g(E(X)).
Ifgis strictly convex, then there exists a line going through the point E(X), say ‘ðxị ẳaỵbx, such that g(x)> ‘ðxị ẳaỵbx 8x∈Ifor whichx6ẳE(X), and g(E(X)) ẳa+ bE(X). Then, assuming that no element inR(X) is assigned proba- bility one, (i.e., X is not degenerate), the previous inequality results become strict, implying E(g(X))>g(E(X)) in either the discrete or continuous cases. n 3.12.3 Necessary and Sufficient Conditions for Positive Semidefiniteness
To Prove that the symmetric matrix Aẳ a11 a12
a21 a22
is positive semidefinite iff a11 0,a22 0, and a11a22 a12a21 0, note that the matrixAwill be positive semidefiniteiffthe characteristic roots ofAare nonnegative (e.g., F.A.
y
x y = g(x)
y = a+bx
E(X) Figure 3.12
Convex functiong.
20Recall the integral inequality that ifh(x) t(x)8x∈(a,b), then Rb
ahðxịdxRb
atðxịdx. Strict inequality holds ifh(x)>t(x)8x∈(a,b).
The result holds foraẳ 1and/orbẳ 1.
164 Chapter 3 Expectations and Moments of Random Variables
Graybill (1983) Matrices with Applications in Statistics, Belmont, CA:
Wadsworth, p. 397). The characteristic roots of A are found by solving the determinantal equation a11l a12
a21 a22l
ẳ0 for l, which can be represented as (a11l)(a22l)a12a21ẳ0 orl2(a11+a22)l ỵ(a11a22 a12a21)ẳ0.
Solutions to this equation can be found by employing the quadratic formula21 to obtain
lẳða11ỵa22ị ỵ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða11ỵa22 ị24ða11 a22a12a21ị q
2
Forlto be 0, it must be the case that the numerator term is 0. Note the term under the square root sign must be nonnegative, since it can be rewritten as (a11 a22)2+ 4a12a21, and becausea12 ẳa21(by symmetry ofA), two nonnega- tive numbers are being added together. If (a11+ a22)>0, thenl 0 only ifa11
a22 a12 a21 0, since otherwise the square root term would be larger than (a11 +a22), and when subtracted from (a11+ a22), would result in a negativel.
Also, the term (a11+ a22) cannot be negative or else at least one of the solutions for l would necessarily be negative. Furthermore, both a11 and a22 must be nonnegative, for if one were negative, then there is no value for the other that would result in both solutions forl being positive.Thus, necessity is proved.
Sufficiency follows immediately, sincea11 0, a22 0, anda11a22–a12a21
0 imply both solutions oflare 0. n
Keywords, Phrases, and Symbols
niẳ1, Cartesian Product Characteristic function Chebyshev’s inequality
Conditional expectation E(Y|x∈B), E(g(Y)|x∈B), E(Y|xẳb), E(g(Y)|xẳb)
Correlation between two random variablesrXY
Correlation bound
Correlation matrix,Corr(X) Covariance between two random
variablessXYor Cov(X,Y) Covariance bound
Covariance matrix,Cov(X)
Cumulant generating functioncX(t) Cumulants,kr
Degenerate density function
Degenerate random variable Expectation of a function of a
multivariate random variable, E(g(X1,. . .,Xn))
Expectation of a function of a random variable, E (g(X))
Expectation of a matrix of random variables
Expectation of a random variable, E(X)
Iterated expectation theorem Jensen’s Inequality
Kurtosis, Excess Kurtosis Leptokurtic
Marginal MGF, marginal cumulant generating function
Markov’s inequality
Means and variances of linear combinations of random variables
Median, med(X) Mesokurtic
MGF Uniqueness theorem Mode, mode(X)
Moment generating function, MGF MX(t)
Moments of a random variable m, the mean of a random variable mr, therth moment about the mean or
rth central moment
mr,s, the (r,s)th joint moment about the mean
m0r;s, the (r,s)th joint moment about the origin
21Recall that the solutions to the quadratic equationax2+bx+cẳ0 are given byxẳb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b24ac p
2a .
m0r, therth moment about the origin Quantile ofX
Platykurtic
Regression curve ofYonX Regression function ofYonX
Skewed density function Skewed to the left Skewed to the right
Standard deviation of a random variables, or std(X)
Symmetric density function Uncorrelated
Unimodal
Variance of a random variables2, or var (X)
Problems
1. A small domestic manufacturer of television sets places a three-year warranty on its picture tubes. During the warranty period, the manufacturer will replace the television set with a new one if the picture tube fails.
The time in years until picture tube failure can be represented as the outcome of a random variableXwith probability density function
fðxị ẳ:005e:005xIð0;1ịðxị:
The times that picture tubes operate until failure can be viewed as independent random variables. The company sells 100 television sets in a given period.
(a) What is the expected number of television sets that will be replaced due to picture tube failure?
(b) What is the expected operating life of a picture tube?
2. A small rural bank has two branches located in neighboring towns in eastern Washington. The numbers of certificates of deposit that are sold at the branch in Tekoa and the branch in Oakesdale in any given week can be viewed as the outcome of the bivariate random variable (X,Y) having joint probability density function
f xð ;yị ẳ x3y3 3;025 ð ị2
" #
Ið0;1;2;...;10ịðxịIð0;1;2;...;10ịðyị:
(a) Are the random variables independent?
(b) What is the expected number of certificate sales by the Oakesdale Branch?
(c) What is the expected number of combined certificate sales for both branches?
(d) What is the answer to b)given thatTekoa branch sells four certificates?
Potentially helpful result:
Xn
xẳ1
x3ẳn2ðnỵ1ị2
4 :
3. The weekly number of luxury and compact cars sold by “Honest” Abe Smith at the Auto Mart, a local car dealership, can be represented as the outcome of a bivari- ate random variable (X,Y) with the nonzero values of its joint probability density function given by
Y
0 1 2 3 4
X
0 .20 .15 .075 .05 .03 1 .10 .075 .04 .03 .02
2 .05 .03 .02 .01 .01
3 .04 .03 .02 .01 .01
Al receives a base salary of $100/week from the dealer- ship, and also receives a commission of $100 for every compact car sold and $200 for every luxury car sold.
(a) What is the expected value of the weekly commission that Al obtains from selling cars? What is the expected value of his total pay received for selling cars?
(b) What is the expected value of his commission from selling compact cars? What is the expected value of his commission from selling luxury cars?
(c) Given that Al sells four compact cars, what is the expected value of his commission from selling luxury cars?
(d) If 38 percent of Al’s total pay goes to federal and state taxes, what is the expected value of his pay after taxes?
(4) The yield, in bushels per acre, of a certain type of feed grain in the midwest can be represented as the outcome of the random variableYdefined by
Yẳ3x:l30x:k45eU
wherexlandxkare the per acre units of labor and capital utilized in production, andUis a random variable with probability density function given by
fðuị ẳ2e2uIð0;1ịðuị:
166 Chapter 3 Expectations and Moments of Random Variables
The price received for the feed grain is $4/bushel, labor price per unit is $10, and capital price per unit is $15.
(a) What is the expected yield per acre?
(b) What is the expected level of profit per acre if labor and capital are each applied at the rate of 10 units per acre?
(c) Define the levels of input usage that maximize expected profit. What is the expected maximum level of profit?
(d) The acreage can be irrigated at a cost of $125 per acre, in which case the yield per acre is defined by Yẳ5xl:30x:45k eU:
If the producer wishes to maximize expected profit, should she irrigate?
5. The daily price/gallon and quantity sold (measured in millionsof gallons) of a lubricant sold on the wholesale spot market of a major commodity exchange is the out- come of a bivariate random variable (P,Q) having the joint probability density function
fðp;qị ẳ2pepqIẵ:5;1ðpịIð0;1ịðqị:
(a) Define the regression curve ofqonp.
(b) Graph the regression curve that you have defined in (a).
(c) What is the expected value of the quantity of lubri- cant sold, given that price is equal to $.75 per gallon?
(d) What is the expected value of total dollar sales of lubricant on a given day?
6. The short-run production function for a particular agricultural crop is critically dependent on the level of rainfall during the growing season, the relationship being Yẳ30ỵ3X.075X2, whereyis yield per acre in bushels, andxis inches of rainfall during the growing season.
(a) If the expected value of rainfall is 20 inches, can the expected value of yield per acre be as high as 70 bushels per acre? Why or why not?
(b) Suppose the variance of rainfall is 40 square inches.
What is the expected value of yield per acre? How does this compare to the bound placed on E(Y) by Jensen’s inequality?
7. For each of the densities below, indicate whether the mean and variance of the associated random variable
exist. In addition, find the median and mode, and indicate whether or not each density is symmetric.
(a)f(x)ẳ3x2I[0, 1](x) (b)f(x)ẳ2x3I[1,1](x)
(c)f(x)ẳ[p(1 +x2)]1I(1,1)(x) (d)fðxị ẳ 4
x
:2
ð ịxð ị:8 4xIð0;1;2;3;4ịðxị
8. The daily price of a certain penny stock is a random variable with an expected value of $2. Then the probabil- ity is.20 that the stock price will be greater than or equation to $10. True or false?
9. The miles per gallon attained by purchasers of a line of pickup trucks manufactured in Detroit are outcomes of a random variable with a mean of 17 miles per gallon and a standard deviation of .25 miles per gallon. How probable is the event that a purchaser attains between 16 and 18 miles per gallon with this line of truck?
10. The daily quantity of water demanded by the population of a large northeastern city in the summer months is the outcome of a random variable, X, measured in millions of gallons and having a MGF of Mx(t)ẳ(1.5t)10fort<2.
(a) Find the mean and variance of the daily quantity of water demanded.
(b) Is the density function of water quantity demanded symmetric?
11. The annual return per dollar for two different invest- ment instruments is the outcome of a bivariate random variable (X1,X2) with joint moment-generating function Mx(t)ẳexp(u0tỵ:5t0St), where
tẳ t1
t2 ;uẳ :07 :11
andSẳ :225103 :3103 :3103 :625103
: (a) Find the mean annual return per dollar for each of the
projects.
(b) Find the covariance matrix of (X1,X2).
(c) Find the correlation matrix of (X1, X2). Do the outcomes ofX1and X2 satisfy a linear relationship x1ẳa1+a2x2?
(d) If an investor wishes to invest $1,000 in a way that maximizes her expected dollar return on the invest- ment, how should she distribute her investment
dollars between the two projects? What is the vari- ance of dollar return on this investment portfolio?
(e) Suppose the investor wants to minimize the variance of her dollar return. How should she distribute the
$1,000? What is the expected dollar return on this investment portfolio?
(f) Suppose the investor’s utility function with respect to her investment portfolio isU(M)ẳ5Mb, whereM is the dollar return on her investment of $1,000. The investor’s objective is to maximize the expected value of her utility. If bẳ1, define the optimal investment portfolio.
(g) Repeat (f), but letbẳ2.
(h) Interpret the investment behavior differences in (f) and (g) in terms of investor attitude toward risk.
12. Stanley Statistics, an infamous statistician, wants you to enter a friendly wager with him. For $1,000, he will let you play the following game. He will continue to toss a fair coin until the first head appears. Letting x represent the number of times the coin was tossed to get the first heads, Stanley will then pay you $2x.
(a) Define a probability space for the experiment of observing how many times a coin must be tossed in order to observe the first heads.
(b) What is the expected payment that you will receive if you play the game?
(c) Do you want to play the game? Why or why not?
13. The city of Megalopolis operates three sewage treat- ment plants in three different locations throughout the city. The daily proportion of operating capacity exhibited by the three plants can be represented as the outcome of a trivariate random variable with the following probability density function:
fðx1;x2;x3ị ẳ1
3ðx1ỵ2x2ỵ3x3ịP3
iẳ1Ið0;1ịðxiị;
wherexiis the proportion of operating capacity exhibited by planti,iẳ1, 2, 3.
(a) What are the expected values of the capacity proportions for the three plants, i.e., what is E
X1
X2 X3
2 4
3 5?
(b) What is the expected value of the average proportion of operating capacity across all three plants, i.e., what is E 13P3
iẳ1Xi
?
(c)Giventhat plant 3 operates at 90 percent of capacity, what are the expected values of the proportions of capacity for plants 1 and 2?
(d) If the daily capacities of plants 1 and 2 are 100,000 gal of sewage each, and if the capacity of plant three is 250,000 gal, then what is the expected daily number of gallons of sewage treated by the city of Megalopolis?
14. The average price and total quantity sold of an econ- omy brand of ballpoint pen in a large western retail mar- ket during a given sales period is represented by the outcome of a bivariate random variable having a probabil- ity density function
fðp;sị ẳ10peps Iẵ:10; :20ðpịIð0;1ịðsị
wherepis the average price, in dollars, of a single pen and sis total quantity sold, measured in 10,000-pen units.
(a) Define the regression curve ofSonP.
(b) What is the expected quantity of pens sold, given that price is equal to $0.12? (You may use the regression curve if you wish.)
(c) What is the expected value of total revenue from the sale of ball point pens during the given sales period, i.e., what is E(PS)?
15. A game of chance is considered to be “equitable” or
“fair” if a player’s expected payoff is equal to zero. Exam- ine the following games:
(a) The player rolls a pair of fair dice. LetZrepresent the amount of money that the player lets on the game outcome. If the player rolls a 7 or 11, the player payoff is 2Z (i.e., he gets to keep his bet of $Z, plus he receives an additional $2Z). If the player does not roll a 7 or 11, he loses the $Z that he bet on the game. Is the game fair?
(b) The player spins a spinner contained within a disk, , that is segmented into five pieces as
A
B C D
E 168 Chapter 3 Expectations and Moments of Random Variables
where P(A)ẳ1/3, P(B)ẳ1/6, P(C)ẳ1/6, P(D)ẳ1/12, P(E)ẳ1/4.
Each spin costs $1. The payoffs corresponding to when the spinner lands in one of the five segments are given by:
Segment Payoff
A $.60
B $1.20
C $1.20
D $2.40
E $.80
Is the game fair?
(c) A fair coin will be tossed repeatedly until heads occurs. If the heads occurs on the jth toss of the coin, the player will receive $2j. How much should the player be charged to play the game if the game is to be fair? (Note: This is a trick question and represents the famous “St. Petersburg paradox” in the statistical literature.)
16. The manager of a bakery is considering how many chocolate cakes to bake on any given day. The manager knows that the number of chocolate cakes that will be demanded by customers on any given day is a random variable whose probability density is given by
fðxị ẳxỵ1
15 If0;1;2;3gðxị ỵ7x
15 If4;5gðxị:
The bakery makes a profit of $1.50 on each cake that is sold. If a cake is not sold on a given day, the cake is thrown away (because of lack of freshness), and the bakery loses
$1. If the manager wants to maximize expected daily profit from the sale of chocolate cakes, how many cakes should be baked?
17. The daily price and quantity sold of wheat in a North- western market during the first month of the marketing year is the outcome of a bivariate random variable (P,Q) having the probability density function
f pð ;qị ẳ:5pepqIẵ3;5ðpịIð0;1ịðqị
wherep is measured in $/bushel, andq is measured in units of 100,000 bushels.
(a) Define the conditional expectation of quantity sold as a function of price, i.e., define E(Q|p) (the regression curve ofQonP).
(b) Graph the regression curve you derived in (a). Calcu- late the values of E(Q|pẳ3.50) and E(Q|pẳ4.50).
18. In each case below, calculate the expected value of the random variableY:
(a) EðYjxị ẳ2x2ỵ3;fXðxị ẳexIð0;1ịðxị:
(b) EðYjxị ẳ3x1x2;Eð ị ẳX1 5;Eð ị ẳX2 7;X1 and X2are independent.
19. The total daily dollar sales in the ACME supermarket is represented by the outcome of the random variableS having a mean of 20, wheresis measured in thousands of dollars.
(a) The store manager tells you the probability that sales will exceed $30,000 on any given day is .75. Do you believe her?
(b) You are now given the information that the variance of the random variableSis equal to 1.96. How proba- ble is the sales event that s∈(10,30)?
20. The first three moments about the origin for the random variableYare given as follows:m01ẳ:5;m02ẳ:5;m03
ẳ:75.
(a) Define the first three moments about themeanforY.
(b) Is the density ofYskewed? Why or why not?
21. The random variableYhas the PDFf(y)ẳy2I[1,1)(y).
(a) Find the mean ofY.
(b) Can you find the first100moments about the origin (i.e.,m01;m02;. . .;m0100) for the random variableY, why or why not?
22. The moment-generating function of the random var- iableYis given byMYðtị ẳð1:25tị3fort<4.
(a) Find the mean and variance of the random variableY.
(b) Is the PDF ofYskewed? Why or why not?
(c) It is known that the moment generating function of the PDF fðxị ẳbaG1ð ịa xa1ex=bIð0;1ịðxị is given by Mxðtị ẳð1btịafort<b1. TheG(a) in the preced- ing expression for the pdf is known as thegamma function, which for integer values ofais such that G(a)ẳ(a1)!. Define the exact functional form of the probability density function forY, if you can.
23. A gas station sells regular and premium fuel. The two storage tanks holding the two types of gasoline are refilled every week. The proportions of the available supplies of regular and premium gasoline that are sold during a given week in the summer is an outcome of a bivariate random variable having the joint density function
f xð ;yị ẳ25ð3xỵ2yịIẵ0;1ðxịIẵ0;1ðyị; where xẳproportion of regular fuel sold andyẳproportion of premium fuel sold.
(a) Find the marginal density function ofX. What is the probability that greater than 75 percent of the avail- able supply of regular fuel is sold in a given week?
(b) Define the regression curve ofYonX, i.e., define E(Y| x). What is the expected value of Y, given that xẳ.75? AreYandXindependent random variables?
(c) Regular gasoline sells for $1.25/gal and premium gas- oline sells for $1.40/gal. Each storage tank holds 1,000 gal of gasoline. What is the expected revenue generated by the sale of gasoline during a week in the summer,giventhatxẳ.75?
24. Scott Willard, a famous weatherman on national TV, states that the temperature on a typical late fall day in the upper midwest, measured in terms of both the Celsius and Fahrenheit scales, can be represented as the outcome of the bivariate random variable (C,F) such that
E C F
" #
ẳ 5 41
andCovðC;Fị ẳ 25 45 45 81
: (a) What is the correlation betweenCandF?
(b) To what extent is there a linear relationship between CandF? Define the appropriate linear relationship if it exists.
(c) Is (C, F) a degenerate bivariate random variable? Is this a realistic result? Why or why not?
25. A fruit processing firm is introducing a new fruit drink, “Peach Passion,” into the domestic market. The firm faces uncertain output prices in the initial marketing period and intends to make a short-run decision by choosing the level of production that maximize the expected value of utility:
EðUðpịị ẳEð ị p avarðpị:
Profit is defined bypẳPqCðqị;pis the price received for a unit of Peach Passion,Uis utility, the cost function is defined byc(q)ẳ.5q2,a0 is a risk aversion parame- ter, and the probability density function of the uncertain output price is given byf(p)ẳ.048(5pp2)I[0,5](p).
(a) If the firm were risk neutral, i.e.,aẳ0, find the level of production that maximizes expected utility.
(b) Now consider the case where the firm is risk averse, i.e.,a>0. Graph the relationship between the opti- mal level of output and the level of risk aversion (i.e., the level ofa). How large doesahave to be for optimal qẳ1?
(c) Assume that aẳ1. Suppose that the Dept. of Agri- culture were toguaranteea price to the firm. What guaranteedprice would induce the firm to produce the same level of output as in the case where price was uncertain?
26. A Seattle newspaper intends to administer two different surveys relating to two different anti-tax initiatives on the ballot in November. The proportion of surveys mailed that will actually be completed and returned to the newspaper can be represented as the out- come of a bivariate random variable (X,Y) having the density function
f xð ;yị ẳ2
3ðxỵ2yịIẵ0;1ðxịIẵ0;1ðyị;
wherexis the proportion of surveys relating to initiative I that are returned, andyrefers to the proportion of surveys relating to initiative II that are returned.
(a) AreXandYindependent random variables?
(b) What is the conditional distribution of x, given yẳ.50? What is the probability that less than 50 percent of the initiative I surveys are returned, given that 50 percent of the initiative II surveys are returned?
(c) Define the regression curve of X on Y. Graph the regression curve. What is the expected proportion of initiative I surveys returned, given that 50 percent of the initiative II surveys are returned?
27. An automobile dealership sells two types of four-door sedans, the “Land Yacht” and the “Mini-Rover.” The number of Land Yachts and Mini-Rovers sold on any given day varies, with the probabilities of the various possible sales outcomes given by the following table:
Number of Mini-Rovers sold
Number of Land Yachts sold
0 1 2 3
0 .05 .05 .02 .02
1 .03 .10 .08 .03
2 .02 .15 .15 .04
3 .01 .10 .10 .05
170 Chapter 3 Expectations and Moments of Random Variables