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Homework 4: Part A Due Friday, Feb 14 During Sections ECON 451 Intermediate Introduction to Stats. and Econometrics I Winter 2020 Homework 4 Part A covers material from Lecture 6 and Lecture 7. Part B will be released at later date but contains much less questions. Note: Grades are based on completion and sufficient effort. Insufficient effort will receive point deductions. Most questions are taken from Larsen and Marx with additional outside source material. A. Continuous Probability Distribution Functions 1. For a random variable Y, given fY (y) calculate the following probabilities, (a) P (0 ≤ Y ≤ 12 ) if fY (y) = 4y 3 for 0 ≤ y ≤ 1 (b) P ( 34 ≤ Y ≤ 1) if fY (y) = 2 3 + 2 3y for 0 ≤ y ≤ 1 (c) P (|Y − 12 |< 1="" 4="" )="" if="" fy="" (y)="3" 2y="" 2="" for="" −1="" ≤="" y="" ≤="" 1="" (d)="" p="" (y="">< 2)="" if="" fy="1" y="" for="" 1="" ≤="" y="" ≤="" e="" 2.="" in="" a="" certain="" country,="" the="" distribution="" of="" a="" family’s="" disposable="" income,="" y,="" is="" described="" by="" the="" pdf="" fy="" (y)="ye" −y,="" y="" ≥="" 0.="" find="" fy="" (y).="" 3.="" suppose="" a="" function="" f(y)="ay2" for="" 0="" ≤="" y="" ≤="" 2="" and="" f(y)="0" elsewhere.="" (a)="" find="" a="" value="" of="" a="" such="" that="" the="" function="" is="" a="" valid="" pdf.="" (b)="" calculate="" p="" (1="" ≤="" y="" ≤="" 2).="" (c)="" find="" the="" cdf="" fy="" (y).="" (d)="" graph="" fy="" (y)="" and="" fy="" (y).="" 4.="" show="" that="" the="" following="" fy="" (y)="" is="" a="" valid="" pdf.="" (a)="" let="" n="" be="" a="" positive="" integer.="" fy="" (y)="(n+" 2)(n+="" 1)y="" n(1−="" y)="" for="" 0="" ≤="" y="" ≤="" 1.="" (b)="" fy="" (y)="b" y2="" for="" y="" ≥="" b="" ≥="" 0="" b.="" expected="" values,="" median,="" and="" variance="" 1.="" calculate="" e(y)="" for="" the="" following="" pdfs:="" (a)="" fy="3(1−" y)2="" for="" 0="" ≤="" y="" ≤="" 1="" (b)="" fy="4ye" −2y="" for="" y=""> 0 (c) fY = 3 4 for 0 ≤ y ≤ 1, fY = 1 4 for 2 ≤ y ≤ 3, and fY = 0 elsewhere. (d) fY = sin y for 0 ≤ y ≤ π2 2. Define E[X] = µ which is just a constant. Prove that the following statement concerning the variance of the random variable X is true assuming that E[X2] is finite1. E[(X − µ)2] = E[X2]− µ2 1In other words, it exists. 4-1 4-2 Lecture 4: Part A Due Friday, Feb 14 During Sections 3. Let X be a geometric distribution with p as the probability of success and k is the number of trials until the first success occur. Show that E[X] = 1p . (Hint: You will need to use the identity ∑∞ x=1 x ∗ rx−1 = 1 (1−r)2 for r between 0 and 1 and it will make this proof easy.) 4. In a game of redball, two drawings are made without replacement from a bowl that has four white pingpong balls and two red ping-pong balls. The amount won is determined by how many of the red balls are selected. For a 5 dollar bet, a player can be opt to be paid under rule A or rule B, as shown in the table below. If you were playing the game, which would you choose? Why? (Hint: make your argument with definition of expected values.) A B No. of Red Balls Drawn Payoff No. Of Red Ball Drawn Payoff 0 0 0 0 1 $2 1 $1 2 $10 2 $20 5. Find the Var of Y if fY (y) = 3 4 , 0 ≤ y ≤ 1 1 4 , 2 ≤ y ≤ 3 0 elsewhere 6. Random variable X has a poisson distribution. This distribution is special because its parameter λ is both the mean and the variance of X. Without needing to know the pdf, calculate the following if we know that λ = 2 (thus mean and variance is 2). (a) E[X2] (b) If W = 50− 2X −X2, what is E[W]? 7. Find the median for each of the following pdfs: (a) fY (y) = (θ + 1)y θ for 0 ≤ y ≤ 1, θ > 0 (b) fY (y) = y + 1 2 for 0 ≤ y ≤ 1 Homework 4: Part B Due Friday, Feb 14 During Sections ECON 451 Intermediate Introduction to Stats. and Econometrics I Winter 2020 Homework 4 Part B covers material from Lecture 8 and Lecture 9. 1. Let X and Y be two continuous random variable with a valid joint pdf. Find c given the information below. (a) fX,Y = c(x 2 + y2) for 0 ≤ y ≤ 1 and 0 ≤ x ≤ 1 (b) fX,Y = c(x + y) for 0 < x="">< y="">< 1="" 2.="" given="" a="" continuous="" random="" variable="" x="" and="" y,="" calculate:="" (a)="" fx,y="" (x,="" y)="" if="" fx,y="" (x,="" y)="1" 2="" and="" 0="" ≤="" x="" ≤="" y="" ≤="" 2="" (b)="" fx,y="" (x,="" y)="" if="" fx,y="" (x,="" y)="1" x="" and="" 0="" ≤="" y="" ≤="" x="" ≤="" 1="" added="" hint:="" to="" solve="" the="" question="" above,="" you="" need="" to="" use="" integration="" by="" parts="" when="" you="" see∫="" ln="" y.="" you="" will="" also="" need="" to="" use="" l’hospital’s="" rule="" when="" you="" see="" limy→0="" y="" ln="" y="" (c)="" fx(x)="" and="" fy="" (y)="" if="" fx,y="" (x,="" y)="2" 3="" (x="" +="" 2y)="" and="" 0="" ≤="" x="" ≤="" 1,="" 0="" ≤="" y="" ≤="" 1="" (d)="" fx(x)="" and="" fy="" (y)="" if="" fx,y="" (x,="" y)="6x" and="" 0="" ≤="" x="" ≤="" 1,="" 0="" ≤="" y="" ≤="" 1−="" x="" 3.="" an="" urn="" contains="" twelve="" chips="" -="" four="" red,="" three="" black,="" and="" five="" white.="" a="" sample="" size="" 4="" is="" to="" be="" drawn="" without="" replacement.="" let="" x="" denote="" the="" number="" of="" white="" chips="" in="" the="" sample="" and="" y="" is="" the="" number="" of="" red.="" find="" fx,y="" (1,="" 2).="" 4.="" given="" the="" joint="" pdf="" fx,y="" (x,="" y)="2e" −(x+y),="" 0="" ≤="" x="" ≤="" y,="" y="" ≥="" 0="" (4.1)="" (a)="" p="" (y="">< 1|x="">< 1)="" (b)="" p="" (y="">< 1|x="1)" (c)="" fy="" |x(y)="" (d)="" e(y="" |x)="" 5.="" suppose="" that="" random="" variables="" x="" and="" y="" are="" independent="" with="" marginal="" pdfs="" fx(x)="2x," 0="" ≤="" x="" ≤="" 1="" and="" fy="" (y)="3y" 2,="" 0="" ≤="" y="" ≤="" 1.="" find="" p="" (y="">< x). 4-1 x).="">