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STA 313 Assignment

创建日期：2024-05-20 16:19:42

所属分类：统计学statistics

标签： STA 313

Assignment

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STA 313

Some of these problems will be marked and posted on Crowdmark.
Problem 1 Let X be a r.v. having a normal distribution, N (a, σ2) with the parameters
EX = a and E(X − Ex)2 = σ2 > 0. Find P(X > a). Compute E (X −K)+, K > a.
Problem 2 Let ξ1, ξ2, ξn be i.i.d. random variables having uniform distribution in the in-
terval [0, 1]. Find the limit
lim
n→∞
(ξ1 · ξ2 · . . . · ξn)1/n .
Solve this problem using Monte Carlo method and analytically, using the Law of Large Num-
bers.
Problem 3 Let ξ1, ξ2, . . . ξn be i.i.d. random variables with mean 0 and finite variance,
0 < σ2 <∞. Prove that
n∑
k=1
ξk√
n∑
k=1
ξ2k
→ ξ in distribution as n→∞ ,
1
where ξ is a random variable having standard normal distribution.
Problem 4 Let X be a random variable, f and g are monotonically increasing, bounded
functions. Prove that f(X) and g(X) are positively correlated.
Hint: consider E
[
(f(X)− f(X ′)) · (g(X)− g(X ′))
]
, where X and X ′ are i.i.d.
Problem 5 (!) Let X and Y be r.v. such that EX2 < ∞ and EY 2 < ∞. Suppose
E(X |Y ) = Y and E(Y |X) = X. Prove that Y = X with probability 1.
Hint: compute E(X − Y )2 using conditioning on X and Y .
Problem 6 Let X be a normal N (0, 1) r.v.
i. Prove that
P(X > t) ≤ 1√
2pit
exp
(
−t
2
2
)
, t > 0.
ii. Compute E[Φ(aX + b)], where, as usual,
Φ(x) =
1√
2pi
∫ x
−∞
exp
(
−t
2
2
)
dt, a, b ∈ R.
Problem 7 Let X be a random variable having Poisson distribution with parameter λ > 0:
P (X = k) = e−λ
λk
k !
, k = 0, 1, 2, . . . .
Introduce a random variable Y such that Y = 0 if X is odd and Y = 1 if X is even. Find
E[X |Y ].
Problem 8 (Wald’s Identity) Let Xk be i.i.d. random variables such that E[X1] = m <
∞. Denote Sn =
∑n
k=1Xk. Let τ be an integer random variable independent of Xk. Prove
that
E
[ τ∑
k=1
Xk
]
= m · E[τ ].
Problem 9 Let Xn be i.i.d. with EXn = 0. Consider Sn =
n∑
k=1
Xk. Let τ ≥ 1 be a random
integer with finite mean, Eτ <∞. Prove that if σ2(Xn) <∞ then
σ2(Sτ ) = σ
2(X1) · Eτ.

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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