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Introduction to Sampling / Convergence and Asymptotic theorems Flashcards | StudBud
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Introduction to Sampling / Convergence and Asymptotic theorems
Introduction to Sampling / Convergence and Asymptotic theorems
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Answers
1
In theory, to ensure that the X
i
are iid, we need to draw the sample randomly and without replacement.
True
False
2
Let (X
1
,..., X
n
) be a
n
-sample of
X
, the CLT implies that \overline{X_n}is asymptotically a Standard Normal
True
False
3
Let (X
1
,..., X
n
) be a
n
-sample of distribution X, what is the limit in Probability of \frac{1}{n}\sum_{i=1}^nX
i
3
?
E(X3)
X
X3
E(X)
4
We collect a sample of the random variable X. Then (X
1
, ... , X
n
) is called a n-sample, if
Xi are iid
Xi are independent
Xi follow the same distribution
Xi are correlated
5
If we survey n individuals from a population of size N, then what is the
sampling rate
?
N
N/n
n/N
n
6
We roll a die 3 times and denote the outcome as X
1
, X
2
, X
3
. Then (X
1
, X
2
, X
3
) is a 3-sample of which
distribution
?
Geometric(1/3)
Uniform on {1,2,3,4,5,6}
Binomial(3, 1/3)
Bernoulli(1/3)
7
If we toss a coin twice and define X
1
, X
2
the resulting 2 random variables, then (X
1
,X
2
) is a iid 2-sample of a Ber(0.5)
False
True
8
Let (X
1
,..., X
n
) be a
n
-sample of distribution X, then \sum_{i=1}^nX
i
and
n.
X follow the same distribution.
True
False
9
What are the assumptions of the CLT ?
(X1,...Xn) iid and V(X)<∞
(X1,...Xn) iid and E(X) <∞
(X1,...Xn) a n-sample, E(X) and V(X) exist
(X1,...Xn) a n-sample
10
Let (X
1
,..., X
n
) be a
n
-sample of Exp(λ) and
T
n
=\frac{1}{\overline{X_n}}, what is the limit in Probability of
T
n
?
λ^{^{ }}
e^{-λ}
e^λ
1 /λ
11
Contrary to Descriptive Statistics, Inferential Statistics deal with
Partial data
Simulated data
Real data
Full data
12
The empirical frequency is the special case of \overline{X_n}for a Bernoulli
n
-sample.
True
False
13
If
X
~
N
(
m
,σ) then (
X-m
)/σ^2~
N
(0,1)
True
False
14
Convergence in Probability⇒Convergence in Distribution
False
True
15
If (X
1
, ... , X
5
) is a 5-sample from Standard Normal distribution, what is the
expected sample mean
?
1/5
1
5
0
16
The Moivre-Laplace theorem is an application of the CLT on a Bernoulli
n
-sample and says that:
B(n,p)→N(0,p(1-p))
B(n,p)→N(p,\sqrt{\frac{p\left(1-p\right)}{n}})
B(n,p)→N(np,\sqrt{np\left(1-p\right)})
B(n,p)→N(0,1)
17
T=g(X
1
, ... , X
n
) is called a sample statistics. T is also a random variable.
True
False
18
What theorems did you use to find the limit in Probability of T
n
in the previous question?
CLT and Slutsky
LLN and CLT
LLN and Mapping Theorem
LLN and Slutsky
