pls complete this pset thanks
ECON 3130 DEPARTMENT OF ECONOMICS INTRODUCTION TO PROBABILITY AND STATISTICS CORNELL UNIVERSITY FALL 2020 Problem Set #2 Due: September 30, 8:00am You may work with your study group on this assignment. However, even if you work with others you must write up and submit your own assignment. Late assignments will not be accepted. 1. SupposeX is a continuous random variable, with density f given by f(t) = 0 if t ≤ −1 or t ≥ 1, and f(t) = .5(1 + t) if t ∈ (−1, 1). (a) Show that f is a proper density. In particular, show that it is non- negative at all evaluation points and integrates to one. (b) Graph f . (c) Write down the equation for FX . Graph FX . (d) Check that the derivative of cdf FX is f at the points where f is contin- uous. (e) Compute the following probabilities: i. Pr[X ∈ (0, 1/2)] ii. Pr[X ∈ (0, 1/2]] iii. Pr[X ∈ (0, 1/2) ⋃ (−1, 1/2)] iv. Pr[X ∈ (0, 1/2) ⋃ (1/4, 3/4)] 2. Answer true or false to the following statements, and provide a justification. (a) For any random variable X , E(X4) ≥ [E(X2)]2. (b) For any random variable X , Var(X2) ≥Var(X). (c) If X ∼Uniform[0, 2], and Y = 1[X ∈ (.5, 1)], then Var(Y ) = 3/16. (d) If Y = 1[X ∈ A], Z = 1[X ∈ B], with A ⋂ B = ∅, then E(Y ) + E(Z) = Pr[X ∈ A ⋃ B]. 3. Suppose that X is a random variable that has a normal distribution with mean µ = 5 and standard deviation σ = 10. Evaluate the following probabilities: 1 (a) Pr(X ≥ 10) (b) Pr(X < 2)="" (c)="" pr(6="" ≤="" x="" ≤="" 11)="" (d)="" pr((x="" −="" 10)2="" ≤="" 12)="" 4.="" the="" waiting="" time,="" in="" hours,="" between="" successive="" speeders="" spotted="" by="" a="" radar="" unit="" is="" a="" continuous="" random="" variable,="" x;="" with="" cumulative="" distribution="" func-="" tion="" f="" (x)="{" 0,="" for="" x="" ≤="" 0="" δ="" (1−="" e−3πx)="" ,="" for="" x=""> 0 where δ is a constant. (a) What must the value of δ be? Why? (b) Compute the probability that the police will wait less than 7 minutes between successive speeders. (c) Compute E(X) (d) Compute V ar(X) 5. Prove that the skewness of a uniform random variable (ranging from α to β) is 0. It is not enough to just say it must be zero because the uniform is symmetric. I want you to actually compute the skewness. 6. Suppose that on average, on Friday night, a privately owned liquor store serves 50 customers in a 1 and a half hour time period. Assume the customers arrive as part of a Poisson process. (a) What is the probability that the store will serve more than 35 customers in a particular 1 and a half hour time period? 1 (b) On Friday night, over a 4 and a half hour time period, how many cus- tomers should we expect to be served in the liquor store? 7. For the gamma distribution, prove that (a) E(X) = αθ (b) V ar(X) = αθ2 8. Find the value of z for a standard normal random variable Z such that: (a) Pr(Z < z)="0.7580" 2="" (b)="" pr(z=""> z) = 0.7580 (c) Pr(−z < z="">< z)="0.9900" (d)="" pr(0="">< z="">< z) = 0.4875 9. if a large set of observations is normally distributed, approximately what percentage of the observations will differ from the mean by: (a) more than 1.5 σ (b) less than 0.27 σ 10. suppose that returning from spring break, five students are at ithaca’s air- port, waiting in line for taxicabs to go back to their dorms/apartments. sup- pose that the five students don’t know each other and that they will take separate cabs. the first person in line will take the first cab that arrives, the second person will take the second cab that arrives, and so on (the fifth per- son will take the fifth cab that arrives). suppose cabs arrive according to a poisson process, and that during a 30 minute time interval the expected number of cabs is 5. suppose the five people line up at 3:00pm. (a) what is the probability that the fifth person gets a cab by 3:30pm? why? (b) what is the probability that all five people get a cab by 3:30pm? why? (c) suppose that by 3:12pm the first and the second person (but not the third) get a cab. what is the probability that the remaining three people get cabs by 3:30pm? (d) what is the expected number of people (out of five) that will still be waiting for a cab after 15 minutes? why? 3 z)="0.4875" 9.="" if="" a="" large="" set="" of="" observations="" is="" normally="" distributed,="" approximately="" what="" percentage="" of="" the="" observations="" will="" differ="" from="" the="" mean="" by:="" (a)="" more="" than="" 1.5="" σ="" (b)="" less="" than="" 0.27="" σ="" 10.="" suppose="" that="" returning="" from="" spring="" break,="" five="" students="" are="" at="" ithaca’s="" air-="" port,="" waiting="" in="" line="" for="" taxicabs="" to="" go="" back="" to="" their="" dorms/apartments.="" sup-="" pose="" that="" the="" five="" students="" don’t="" know="" each="" other="" and="" that="" they="" will="" take="" separate="" cabs.="" the="" first="" person="" in="" line="" will="" take="" the="" first="" cab="" that="" arrives,="" the="" second="" person="" will="" take="" the="" second="" cab="" that="" arrives,="" and="" so="" on="" (the="" fifth="" per-="" son="" will="" take="" the="" fifth="" cab="" that="" arrives).="" suppose="" cabs="" arrive="" according="" to="" a="" poisson="" process,="" and="" that="" during="" a="" 30="" minute="" time="" interval="" the="" expected="" number="" of="" cabs="" is="" 5.="" suppose="" the="" five="" people="" line="" up="" at="" 3:00pm.="" (a)="" what="" is="" the="" probability="" that="" the="" fifth="" person="" gets="" a="" cab="" by="" 3:30pm?="" why?="" (b)="" what="" is="" the="" probability="" that="" all="" five="" people="" get="" a="" cab="" by="" 3:30pm?="" why?="" (c)="" suppose="" that="" by="" 3:12pm="" the="" first="" and="" the="" second="" person="" (but="" not="" the="" third)="" get="" a="" cab.="" what="" is="" the="" probability="" that="" the="" remaining="" three="" people="" get="" cabs="" by="" 3:30pm?="" (d)="" what="" is="" the="" expected="" number="" of="" people="" (out="" of="" five)="" that="" will="" still="" be="" waiting="" for="" a="" cab="" after="" 15="" minutes?="" why?="">