0.24 adults 2. (c) Prove that, regardless of the possible values of and, the covariance between X and Y is equal to zero. Show all work.

Transcription

1 1 A socioeconomic stud analzes two discrete random variables in a certain population of households = number of adult residents and = number of child residents It is found that their joint probabilit mass function (pmf) p(, ) corresponds to a probabilit table having the structure that appears below, for some parameters and 0 0 (These ranges are necessar in order for all the probabilities in the table to be between 0 and 1) 0 1 p ( ) p ( ) 15 ( ) ( ) (a) Indicate the marginal pmf p ( ) on the table, and calculate the mean and variance of Show all work Mean( ) E[ ] p( ) (1)(06) ()(04) 14 adults Var( ) E[ ] E[ ] p ( ) (1 )(06) ( )(04) (14) (b) Indicate the marginal pmf p ( ) on the table, and calculate the mean and variance of Show all work Mean( ) E[ ] p ( ) Var( ) E[ ] E[ ] p ( ) 04 adults (0)( ) (1)(15 ) ()( 05) 05 children (0 )( ) (1 )(15 ) ( )( 05) (05) ( ) 075 children (c) Prove that, regardless of the possible values of and, the covariance between and is equal to zero Show all work Cov(, ) E[ ] E[ ] E[ ] p(, ) (1)(0)( ) (1)(1)(09 ) (1)()( 0) ()(0)( ) ()(1)(06 ) ()()( 0) (14)(05) (09 ) ( 0) (06 ) 4( 0) (14)(05) (09 ) ( 06) (1 4 ) ( 4 08)

2 (d) Despite (c), onl under a specific condition on and are and statisticall independent Find it; show all work In order for and to be independent, each cell probabilit must be equal to the product of its corresponding row and column marginal probabilities Working with the first cell, this translates to the condition that (06)( ), which simplifies to 15 (Technicall, the other cells should also be checked for the same result However, in a table such as this, it is sufficient to check onl one other cell in either of the two remaining columns The reason is subtle, and has to do with the number of degrees of freedom the table has For the purposes of this eam though, checking one cell is enough) (e) Another variable of interest in the stud is the total number of residents per household Calculate the mean and variance of + Show all work [Hint: Use (a),(b), and (c)] Recall that for the mean, E[ ] E[ ] E[ ] = 19 residents, and Var( ) Var( ) Var( ) Cov(, ) 04 [( ) 075] 0 ( ) 051 residents Note: To solve this part, a few people went back to first principles, and eplicitl constructed the sample space of all the possible ordered pairs (, ), where = 1 or adults, and = 0, 1, or children, then a probabilit table for the sum +, from which the mean and variance could be computed via their original epected value definitions That does work, but is a horrendous waste of time, which is wh the formulas above eist Don t make the problem more complicated than necessar (unless ou are asked to do so, eg, problem on Eam ), then complain that the eam was too long (f) Specif the smallest and largest values of the variance of + Eplain Because it is stated at the beginning of the problem that and 0 0, it must follow that the variance ( ) 051 found in (e) is smallest at 049 when 0, 0, and largest at 099, when 045, 0

3 In this problem, ou are asked to prove the epected value propert E[ ] E[ ] E[ ] for an two random variables and For simplicit, assume the are continuous with joint pdf (, ) f Then we have the following definitions: marginal pdf of f ( ) f (, ) d marginal pdf of f ( ) f (, ) d Epected Value of Epected Value of E[ ] f ( ) d E[ ] f ( ) d Epected Value of a real-valued function g(, ) of and E[ g(, )] g(, ) f (, ) d d (a) Use the preceding definition to epress E[ ] as a double integral Soln: Let g(, ) Then E [ ] ( ) f (, ) d d (b) Write this double integral as a sum of two double integrals, and reverse the order of integration of the second from d d to d d over the -plane Soln: E[ ] f (, ) d d f (, ) d d (c) Appl an elementar propert of integrals to rewrite this sum, and epress it in terms of the marginal probabilities of and defined above Briefl eplain E[ ] f (, ) d d f (, ) d d Soln: f ( ) d f ( ) d E[ ] E[ ], b definition The innermost integral in the first term is taken with respect to, keeping fied Hence acts as a constant, and can therefore be passed through that integral smbol Likewise, can be passed through the innermost integral of the second term The rest follows from the definitions (d) Formall justif the conclusion that this is indeed equal to E[ ] E[ ], as claimed

4 Given the set of all uniforml-distributed, random points within a circle of radius 1 It was proved in lecture that the mean distance of these points to the center of the circle is equal to / In this problem, ou are asked to compute the mean distance of these points to an fied point on the boundar of the circle, using a similar method First, without loss of generalit, we ma position the circle so that it is tangent to the -ais from above Furthermore, we ma rotate the circle, so that the fied boundar point is located at the origin (0,0), as shown in the figure The equation of the circle is thus ( 1) 1 For the set of all random, uniforml-distributed points (, ) inside this circle, find the mean distance to the boundar point (0,0) That is, find E (a) Determine the joint pdf f (, ) of the uniform distribution over this circle in the -plane Soln: The area of the circular base is (1), so the uniform pdf must be f (, ) 1 for all points inside the circle (and 0 outside), in order to ensure that the clindrical volume = 1 (b) Prove that the equation of the circle in polar coordinates is range of Soln: ( 1) 1 is equivalent to, ie, r sin, and specif the r r sin, or sin r for 0 (c) Set up BUT DO NOT EVALUATE an epression for the epected distance E Soln: E[ g(, )] g(, ) f (, ) d d, so E d d (d) Evaluate the epression in (b) Note: ou ma use the calculus result cos sin d cos Soln: Switching to polar coordinates, this can be re-epressed as C without proof r sin 0 r 0 1 r dd r dr d sin 1 r sin 1 r 8 r dr d d sin d 0 r r 0 8 cos 8 ( 1) 1 cos ( 1) (= )

5 4 The ages of a population of college undergraduate students are uniforml distributed between and ears old, ie, ~ Unif (, ) (a) Determine the corresponding pdf f( ) for all real values of, and sketch its graph Show all work Solution: As a pdf, the rectangular area under this densit curve of constant height and base width = = 4 must be equal ro 1 Therefore, 1 4, f( ) 0, otherwise (b) Calculate the probabilit that the age of a randoml-selected individual student is between 19 and 1 ears old Show all work Solution: P(19 1) simpl corresponds to the rectangular area of height (ie, densit) = 05, times base width = 1 19 =, or 05 (c) Calculate the mean and variance of the random variable Show all work Solution: The balance point of this smmetric distribution is clearl Or, b definition, 0, b inspection f ( ) d (1 4) d f ( ) d (1 4) d (d) Calculate the (approimate) probabilit that the average age of a sample of n randoml-selected students is between 19 and 1 ears old Show all work Solution: P(19 1) P Z P( 15 Z 15) 4 4 (15) ( 15) = (e) How large would a random sample have to be, in order to reach a 999% probabilit of having its average age lie between 19 and 1 ears old? Show all work Solution: If P( z Z z ) 0999, then ( z) P( Z z ) Therefore, z 9, so n 9 4 n (9) (4 ) 4 n n 144, ie, n 15