Question

Review: Central Limit Theorem

1 point possible (graded)

The **Central Limit Theorem** states that if
X1,…,Xn are i.i.d. and

E[X1]=μ<∞ | ; | Var(X1)=σ2<∞, |

then

n−−√[(1n∑i=1nXi)−μ]−→−−n→∞(d)Wwhere W∼N(0,?). |

What is Var(W)? (Express your answer in terms of n, μ and σ).

Answer #1

What is wrong with the following statement of the central limit
theorem?
Central Limit Theorem. If the random variables X1,
X2, X3, …, Xn are a random sample of size n from any distribution
with finite mean μ and variance σ2, then the distribution of will
be approximately normal, with a standard deviation of σ / √n.

Review: Manipulating Multivariate Gaussians
1 point possible (graded)
Recall that a multivariate Gaussian N(μ⃗ ,Σ) is
a random vector Z=[Z(1),…,Z(n)]T where Z(1),…,Z(n) are
jointly Gaussian , meaning that the density of Z
is given by the joint pdf
f:Rn
→
R
Z
↦
1(2π)n/2det(Σ)−−−−−−√exp(−12(Z−μ⃗ )TΣ−1(Z−μ⃗ ))
where
μ⃗ i
=E[Z(i)],(vector mean).
Σij
=Cov(Z(i),Z(j))(positive definite covariance matrix).
Suppose that Z∼N(0,Σ). Let M denote an n×n matrix.
What is the distribution of MZ?

Confidence interval Concept Check
1 point possible (graded)
As in the previous section, let X1,…,Xn∼iidexp(λ). Let
λˆn:=n∑ni=1Xi
denote an estimator for λ. We know by now that λˆn is a
consistent and asymptotically
normal estimator for λ.
Recall qα/2 denote the 1−α/2 quantile of a standard Gaussian. By
the Delta method:
λ∈[λˆn−qα/2λn−−√,λˆn+qα/2λn−−√]=:I
with probability 1−α. However, I is still not a
confidence interval for λ.
Why is this the case?

Method of Moments Concept Question II
1 point possible (graded)
Let (E,{Pθ}θ∈Θ) denote a statistical model associated to a
statistical experiment X1,…,Xn∼iidPθ∗ where θ∗∈Θ is the true
parameter. Assume that Θ⊂Rd for some d≥1. Let mk(θ):=E[Xk] where
X∼Pθ. mk(θ) is referred to as the k-th moment of
Pθ . Also define the moments map:
ψ:Θ
→Rd
θ
↦(m1(θ),m2(θ),…,md(θ)).
What conditions on ψ do we have to assume so that the method of
moments produces a consistent and asymptotically normal estimator?...

1. The Central Limit Theorem
A. States that the OLS estimator is BLUE
B. states that the mean of the sampling distribution of the
mean is equal to the population mean
C. none of these
D. states that the mean of the sampling distribution of the
mean is equal to the population standard deviation divided by the
square root of the sample size
2. Consider the regression equation Ci= β0+β1 Yi+ ui where C is
consumption and Y is disposable...

(05.02 LC)
The Central Limit Theorem says that when sample size n is taken
from any population with mean μ and standard deviation σ when n is
large, which of the following statements are true? (4 points)
I. The distribution of the sample mean is exactly Normal.
II. The distribution of the sample mean is approximately
Normal.
III. The standard deviation is equal to that of the
population.
IV. The distribution of the population is exactly Normal.
a
I and...

Let X1, . . . , Xn be i.i.d from pmf f(x|λ) where f(x) =
(e^(−λ)*(λ^x))/x!, λ > 0, x = 0, 1, 2
a) Find MoM (Method of Moments) estimator for λ
b) Show that MoM estimator you found in (a) is minimal
sufficient for λ
c) Now we split the sample into two parts, X1, . . . , Xm and
Xm+1, . . . , Xn. Show that ( Sum of Xi from 1 to m, Sum...

1. Let X1, . . . , Xn be i.i.d. continuous RVs with density
pθ(x) = e−(x−θ), x ≥ θ for some unknown θ > 0. Be sure to notice
that x ≥ θ. (This is an example of a shifted Exponential
distribution.) (a) Set up the integral you would solve for find the
population mean (in terms of θ); be sure to specify d[blank]. (You
should set up the integral by hand, but you can use software to
evaluate...

The weak law of large numbers states that the mean of a sample
is a consistent estimator of the mean of the population. That is,
as we increase the sample size, the mean of the sample converges in
probability to the expected value of the distribution that the data
comes from, provided that expected value is finite. Consider a
numerical example, a Student t distribution with n = 5, the same as
we have already seen earlier in this module....

Reconsider the game of roulette. Recall that when you bet $1 on
a color, you have an 18/38 probability of winning $1 and a 20/38
probability of losing $1 (for a net winnings of -$1). Consider
playing for a random sample of n = 4 spins, and consider the
statistic x-bar = sample mean of
your net winnings per spin.
a) Determine the (exact) sampling distribution of
x-bar. [Hint: Start by listing
the possible values of x-bar.
Then use the...

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