PFA
Microsoft Word - STAT 436_HW 5.docx STAT 436: Homework 5, Due: Thursday, 10/21/2021 by 5:00pm 1. Let ? be a random variable with probability mass function ?(?) = ? + 1 6 , ? = 0, 1, 2. a) Derive ?(?), the moment-generating function of ?. b) Use ?(?) to find the mean (?!" ) of the distribution of ?. c) Use ?(?) to find the variance (?#) of the distribution of ?. 2. Let ? be a random variable with probability density function ?(?) = 3?$%& ,0 <>< ∞. a) derive ?(?), the moment-generating function of ?. b) use ?(?) to find the mean (?!" ) of the distribution of ?. c) use ?(?) to find the variance (?#) of the distribution of ?. 3. a manufacturer of tires wants to advertise a mileage interval that excludes no more than 10% of the mileage on tires he sells. all he knows is that, for a large number of tires tested, the mean mileage was 25,000 miles, and the standard deviation was 4000 miles. by chebyshev’s theorem, what interval would you suggest? 4. let the joint probability (mass) function of x and y be given by the following: valuesofx 1 2 valuesofy 1 1/3 1/6 2 1/6 1/3 a) finde(x+y) b) finde(xy) c) findcov(x,y)andcorr(x,y).checkbothofthemarepositive. 5. let ? and ? be two independent random variables. let ? = ?? + ?, and ? = ?? + ?, for some constants ?, ?, ?, and ?. show that ???(?, ?) = 0. 6. let x be a random variable such that p(x ≤ 0) = 0 and let μ = e(x) exists. show that p (x ≥ 2μ) ≤ ! # . ∞.="" a)="" derive="" (?),="" the="" moment-generating="" function="" of="" .="" b)="" use="" (?)="" to="" find="" the="" mean="" (?!"="" )="" of="" the="" distribution="" of="" .="" c)="" use="" (?)="" to="" find="" the="" variance="" (?#)="" of="" the="" distribution="" of="" .="" 3.="" a="" manufacturer="" of="" tires="" wants="" to="" advertise="" a="" mileage="" interval="" that="" excludes="" no="" more="" than="" 10%="" of="" the="" mileage="" on="" tires="" he="" sells.="" all="" he="" knows="" is="" that,="" for="" a="" large="" number="" of="" tires="" tested,="" the="" mean="" mileage="" was="" 25,000="" miles,="" and="" the="" standard="" deviation="" was="" 4000="" miles.="" by="" chebyshev’s="" theorem,="" what="" interval="" would="" you="" suggest?="" 4.="" let="" the="" joint="" probability="" (mass)="" function="" of="" x="" and="" y="" be="" given="" by="" the="" following:="" values="" of="" x="" 1="" 2="" values="" of="" y="" 1="" 1/3="" 1/6="" 2="" 1/6="" 1/3="" a)="" find="" e(x="" +="" y="" )="" b)="" find="" e(xy="" )="" c)="" find="" cov(x,="" y)="" and="" corr(x,="" y="" ).="" check="" both="" of="" them="" are="" positive.="" 5.="" let="" and="" be="" two="" independent="" random="" variables.="" let="" =="" +="" ,="" and="" =="" +="" ,="" for="" some="" constants="" ,="" ,="" ,="" and="" .="" show="" that="" (?,="" )="0." 6.="" let="" x="" be="" a="" random="" variable="" such="" that="" p(x="" ≤="" 0)="0" and="" let="" μ="E(X)" exists.="" show="" that="" p="" (x="" ≥="" 2μ)="" ≤="" !="" #="">