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discrete probability distribution

创建日期：2024-08-26 14:57:08

所属分类：统计学statistics

标签： probability

discrete probability distribution

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Econ-312: Assignment -1

1. Let X be a random variable with a discrete probability distribution with population mean μx . Consider the following relationships:

Variance:

Covariance:

(i) (5 points) Prove that the right hand side of A = right hand side of B. Show all the steps. (ii) (5 points) Prove that the rhs of C = rhs of D. Show all the steps.

2. Consider the following two random variables y1 and y2 , both are linearly related to another random

variable x that has mean ( μx ) and variance ( σx(2) ), as follows:

y1 = a1 + b1x

y2 = a2 + b2x ,

where a1 , b1 , a2 , b2 are constants. For example vacation travel expenditure ( y1 ) may be linearly related to income ( x ) and the number of car ownership ( y2 ) may be linearly related to income ( x ). Based on this information, find the expressions for the following:

(i) (5 points) E(y1 ) =

(ii) (5 points) E(y2 ) =

(iii) (5 points) var(y1 ) =

(iv) (5 points) var(y2 ) =

(v) (5 points) covar(y1, y2 ) =

3. You all know that Unbiasedness and Efficiency are two most important properties of an estimator, which is also often called a sampling statistic. Consider the following two estimators:

where { X1 , X2 , X3 } is a sample of size n = 3 drawn independently from a population with mean (μx) and variance ( σx(2)).

(i) (5 points) Show that both estimators are unbiased.

(ii) (10 points) Prove that the estimator Y1 is more efficient relative to Y2 .

4. Using EXCEL, do the following in sequence:

(i) Generate two random samples ( x and y ) of size 10 from two Normal Distribution – one

from N(μx = 50, σx(2) = 25) and the other from N(μy = 75, σy(2) = 64) . Arrange them in

two columns one with column heading x and the other with column heading y .

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MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 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