statistics

ECON120A: Homework 3
Haitian Xie
Due Wednesday August 24th 11:59 am (noon)
Name (Last, First) PID
For each question, make sure to show all your steps. Simply stating the answer will not
give you full credit.
Exercise 0. Which classmate(s) did you work with?
Exercise 1. [5 points] Warming Up
Suppose I have an i.i.d sample X1, X2, · · · , Xn from the normal distribution distribution with parameters
(µ, σ2
). Recall that these parameters are, respectively, the mean E[X] = µ and the variance of V[X] = σ
2
.
What is the probability that X¯ = µ? Hint: X¯ is a continuous random variable.
Exercise 2. [10 points] Unbiasedness and Consistency
Let X1, · · · , Xn be an i.i.d. sample with mean µ. We have shown that the sample mean estimator X¯
is both unbiased and consistent for the population mean µ. In this question, we study different estimators
that may violate unbiasedness and consistency.
a) Consider the estimator ˆθ1 = X1 for the population mean µ. This is basically using the first observation
as the estimator. Show that this estimator ˆθ1 is unbiased but not consistent.
1
b) Consider the estimator ˆθ2 = X¯ + 1/n for the population mean µ. Show that this ˆθ2 estimator is
consistent but not unbiased.
Exercise 3. [15 points] Sample variance
Suppose we have a sample of i.i.d. random variables X1, X2, · · · Xn from some unknown distribution.
Consider the sample variance estimator ˜σ
2
for the population variance σ
2 = V[Xi
] given by:
σ˜
2
:=
1
n
Xn
i=1
(Xi − X¯)
2
where X¯ is the usual estimator for the sample mean. Show that this estimator is not unbiased. Hint: simply
compute E[˜σ
2
] and show it is different from V[X]. Notice that by using linearity of expectation and the fact
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that Xi have identical distribution we have
E
”
1
n
Xn
i=1
(Xi − X¯)
2
#
=
1
n
Xn
i=1
E[(X1 − X¯)
2
]
= E[(X1 − µ − (X¯ − µ)
2
]
= E[(X1 − µ)
2 − 2(X1 − µ)(X¯ − µ) + (X¯ − µ)
2
]
To finish the proof, which requires a few more steps, use linearity of expectation and the definition of variance.
Notice that, by the definition of variance, E[(X1 − µ)
2
] is equal to the variance of X1, and E[(X¯ − µ)
2
] is
equal to the variance of X¯. Since Xi and Xj are independent for i 6= j, we know that Cov(Xi
, Xj ) = 0.
Exercise 4. [15 points] A gambler’s estimator
X is a random variable with mean E[X] = µ. Consider the following estimator for µ:
Xgambler =
1
n
Xn
i=1
Xi + F (1)
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where F is a random variable that equals to 1 with probability p and −1 with probability (1 − p), and F
independent of Xi for all i.
a) Show that Xgambler is not consistent for µ. Hint: By the marginalization we discussed in Week 2, we have
P(|Xgambler − µ| > ) = P(|Xgambler − µ| > |F = 1) · p + P(|Xgambler − µ| > 0.5|F = −1) · (1 − p)
Choose = 0.5 and explain why the above probability is not converging to zero.
b) For which value of p is Xgambler an unbiased estimator of µ?
c) Compute the variance of Xgambler and compare it to the usual sample mean estimator X¯. Which one
is higher, why?
Exercise 5. [15 points] Sampling Distribution and CLT
Suppose you have access to an i.i.d. sample from the population of people who received a traffic tickets.
We denote Ti as the number of traffic tickets received by person i.
The distribution of each Ti
is given by P(Ti = 0) = 0.8, P(Ti = 1) = 0.15, P(Ti = 2) = 0.05.
a) Compute the E[Ti
] and V[Ti
].
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b) Construct the sampling distribution for the sample mean obtained from a random i.i.d. sample of
n = 2. Remember, the sampling distribution lists out all the possible values for T¯ and the associated probabilities.
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c) Repeat part b) for the the case n = 3.
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d) What is the approximate distribution of the sample mean T¯ when n = 100? Hint: use the central
limit theorem.
Exercise 6. [20 points] Between group inequality
A researcher estimates that, in California, the proportion of people leaving under the State’s poverty line
is equal to p = 0.182. A naive measure of inequality across the two groups is given by:
I(p) := 1 − p
2 − (1 − p)
2 = 2p(1 − p) (2)
a) What is the value of the inequality measure when p = 0. How about p = 1? For which level of p is the
inequality the highest? What about the lowest?
b) Use the continuous mapping theorem to show that I(ˆp)
p→ I(p), that is, the estimated inequality
measure converges to the true inequality measure. Be VERY CAREFUL in showing the assumptions of the
continuous mapping theorem hold in this case.
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c) Compute an estimate for the level of inequality in California based on the information given.
Exercise 7. [20 points] CLT and Emergency Wait Times
Consider an i.i.d. random sample, with n = 100, drawn from the distribution of wait times for an ambulance emergency call is given by the uniform distribution: Xi ∼ U(0, 10)
a) Compute the expected wait time E[Xi
] and the variance of the wait time V[Xi
].
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b) For a specific call Xi
, what is the probability that the wait time is longer than 6 minutes?
c) Use the central limit theorem to construct the approximate distribution of the average wait time X¯
100.
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d) Based on the approximate distribution you give in part c), what is the probability that the average
wait time X¯
100 is higher than 6 minutes? Give an interpretation for why your answer is different from part
b).
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