DTSA 5002 - Statistical Inference for Estimation in Data
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1. Expected Value Property when X and Y are independent, - Correct
Answer: E[XY] = E[X]E[Y]
2. Variance - Correct Answer: Measure of "spread" of a distribution;
standard deviation squared; Var[X] = E[(X-mu)^2] = E[X^2] - (E[X])^2, where
mu = E[X]
3. Properties of Variance - Correct Answer: V(X)>=0, V(aX+b)=a^2V(X)
4. True or False: Var[X+Y] = Var[X] + Var[Y] - Correct Answer: True *only* if
X and Y are independent
5. Covariance - Correct Answer: A measure of linear association between
two variables. Positive values indicate a positive relationship; negative
values indicate a negative relationship; E[XY] - E[X]E[Y]
,6. Correlation - Correct Answer: A measure of the extent to which two
factors vary together, and thus of how well either factor predicts the other.;
Cov(X,Y) / sqrt(Var(x)Var(Y))
7. True or False: if Corr(X,Y) = 0 , then X and Y are uncorrelated. - Correct
Answer: True
8. True or False: A random sample means that it is independent and identically
distributed. - Correct Answer: True
9. True or False: if E[xbar]=mu, then xbar is an unbiased estimator of mu -
Correct Answer: True
10.Moment Generating Function - Correct Answer: M.X(t) = E[e^(tX)]
11.Method of Moments Estimator - Correct Answer: An estimator obtained
by using the sample analog of population moments; ordinary least squares
and two stage least squares are both method of moments estimators
12.maximum likelihood estimation - Correct Answer: a class of estimators
that chooses a set of parameters that provides the highest probability of
observing a particular outcome
13.likelihood function - Correct Answer: joint probability distribution of the
data, treated as a function of the unknown coefficients
,14.Asymptotically unbiased - Correct Answer: unbiased as the sample size
tends to infinity. Some biased estimators are asymptotically unbiased but all
unbiased estimators are asymptotically unbiased.
15.Invariance property of MLEs - Correct Answer: if ^θ is the MLE of θ,
then τ(^θ)τ(θ^) is the MLE of τ(θ).
16.There are some constrains on the choice of τ(θ)τ(θ). If τ(θ)τ(θ) is one-to-one
then this definition is fine. In this case, denote η=τ(θ)η=τ(θ), then the
inverse function τ−1(η)=θτ−1(η)=θ exists
17.Mean Squared Error (MSE) - Correct Answer: the average of the
squared differences between the forecasted and observed values
18.relative efficiency - Correct Answer: given two unbiased point
estimators of the same population parameter, the point estimator with the
smaller standard error is more efficient
19.Cramer Rao Lower Bound - Correct Answer: Var[tau^(theta)] >=
[tau'(theta)] ^2/ I.n(theta), where tau^(theta) is an unbiased estimator
tau(theta)
20.Cauchy-Schwarz Inequality - Correct Answer: If x and y are vectors in Rn,
then |x*y| <= ||x|| ||y||
, 21.UMVUE - Correct Answer: Uniformly Minimum Variance Unbiased
Estimator
22.Weak Law of Large Numbers - Correct Answer: if you take a
representative sample of observations from a population, the mean of that
sample approaches the mean of the population as the sample size increases
23.Markov's Inequality - Correct Answer: P(X>=c)<=E(X)/c
24.Chebyshev's Inequality - Correct Answer: For any distribution with finite
variance, the proportion of the observations within k standard deviations of
the arithmetic mean is at least 1-1/k^2 for all k>1
25.(R code) Probability that at standard normal is <= 'x' - Correct Answer:
pnorm(x)
26.Central Limit Theorem - Correct Answer: As the size n of a simple
random sample increases, the shape of the sampling distribution of x̄ tends
toward being normally distributed.
27.True or False: Consistent estimation means convergence in probability -
Correct Answer: True
28.Asymptotic efficiency - Correct Answer: lim(n-
>inf)(CRLB(theta)/Var[theta^])=1
Science Comprehensive Resource To Help You Ace 2026-
2027 Exams Includes Frequently Tested Questions With
ELABORATED 100% Correct COMPLETE SOLUTIONS
Guaranteed Pass First Attempt!! Current Update!!
Instant Download Pdf
1. Expected Value Property when X and Y are independent, - Correct
Answer: E[XY] = E[X]E[Y]
2. Variance - Correct Answer: Measure of "spread" of a distribution;
standard deviation squared; Var[X] = E[(X-mu)^2] = E[X^2] - (E[X])^2, where
mu = E[X]
3. Properties of Variance - Correct Answer: V(X)>=0, V(aX+b)=a^2V(X)
4. True or False: Var[X+Y] = Var[X] + Var[Y] - Correct Answer: True *only* if
X and Y are independent
5. Covariance - Correct Answer: A measure of linear association between
two variables. Positive values indicate a positive relationship; negative
values indicate a negative relationship; E[XY] - E[X]E[Y]
,6. Correlation - Correct Answer: A measure of the extent to which two
factors vary together, and thus of how well either factor predicts the other.;
Cov(X,Y) / sqrt(Var(x)Var(Y))
7. True or False: if Corr(X,Y) = 0 , then X and Y are uncorrelated. - Correct
Answer: True
8. True or False: A random sample means that it is independent and identically
distributed. - Correct Answer: True
9. True or False: if E[xbar]=mu, then xbar is an unbiased estimator of mu -
Correct Answer: True
10.Moment Generating Function - Correct Answer: M.X(t) = E[e^(tX)]
11.Method of Moments Estimator - Correct Answer: An estimator obtained
by using the sample analog of population moments; ordinary least squares
and two stage least squares are both method of moments estimators
12.maximum likelihood estimation - Correct Answer: a class of estimators
that chooses a set of parameters that provides the highest probability of
observing a particular outcome
13.likelihood function - Correct Answer: joint probability distribution of the
data, treated as a function of the unknown coefficients
,14.Asymptotically unbiased - Correct Answer: unbiased as the sample size
tends to infinity. Some biased estimators are asymptotically unbiased but all
unbiased estimators are asymptotically unbiased.
15.Invariance property of MLEs - Correct Answer: if ^θ is the MLE of θ,
then τ(^θ)τ(θ^) is the MLE of τ(θ).
16.There are some constrains on the choice of τ(θ)τ(θ). If τ(θ)τ(θ) is one-to-one
then this definition is fine. In this case, denote η=τ(θ)η=τ(θ), then the
inverse function τ−1(η)=θτ−1(η)=θ exists
17.Mean Squared Error (MSE) - Correct Answer: the average of the
squared differences between the forecasted and observed values
18.relative efficiency - Correct Answer: given two unbiased point
estimators of the same population parameter, the point estimator with the
smaller standard error is more efficient
19.Cramer Rao Lower Bound - Correct Answer: Var[tau^(theta)] >=
[tau'(theta)] ^2/ I.n(theta), where tau^(theta) is an unbiased estimator
tau(theta)
20.Cauchy-Schwarz Inequality - Correct Answer: If x and y are vectors in Rn,
then |x*y| <= ||x|| ||y||
, 21.UMVUE - Correct Answer: Uniformly Minimum Variance Unbiased
Estimator
22.Weak Law of Large Numbers - Correct Answer: if you take a
representative sample of observations from a population, the mean of that
sample approaches the mean of the population as the sample size increases
23.Markov's Inequality - Correct Answer: P(X>=c)<=E(X)/c
24.Chebyshev's Inequality - Correct Answer: For any distribution with finite
variance, the proportion of the observations within k standard deviations of
the arithmetic mean is at least 1-1/k^2 for all k>1
25.(R code) Probability that at standard normal is <= 'x' - Correct Answer:
pnorm(x)
26.Central Limit Theorem - Correct Answer: As the size n of a simple
random sample increases, the shape of the sampling distribution of x̄ tends
toward being normally distributed.
27.True or False: Consistent estimation means convergence in probability -
Correct Answer: True
28.Asymptotic efficiency - Correct Answer: lim(n-
>inf)(CRLB(theta)/Var[theta^])=1
