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statistical sampling to auditing

Questions and Answers of Statistical Sampling To Auditing

Compare and contrast the notions of chance vs. deterministic regularities.
Explain why the slogan “All models are wrong, but some are useful” conflates two different types of being wrong using the distinction between statistical and substantive inadequacy.
In relation to the experiment of casting two dice (Table 1.3), evaluate the probability of events A – the sum of the two dice is greater than 9 and B – the difference of the two dice is less than
Discuss the connection between observed frequencies and the probabilistic reasoning that accounts for those frequencies.
In relation to the experiment of casting two dice, explain why focusing on (i) adding up the two faces and (ii) odds and evens constitute two different probability models stemming from the same
Explain the connection between a histogram and the corresponding probability distribution using de Mere’s paradox.
Give four examples of variables measured on each of the different scales, beyond those given in the discussion above.
(a) Compare the different scales of measurement.(b) Why do we care about measurement scales in empirical modeling?
Beyond the measurement scales, what features of the observed data are of interest from the empirical modeling viewpoint?
(a) In the context of descriptive statistics, explain briefly the following concepts: (i)mean, (ii) median, (iii) mode, (iv) variance, (v) standard deviation, (vi) covariance,(vii) correlation
Compare and contrast time-series, cross-section, and panel data as they relate to heterogeneity and dependence.
Explain how the different features of observed data can be formalized in the context of expressing a data series in the form {xk, xk ∈ RX, k ∈ N} .
Explain briefly the connection between chance regularity patterns and probability theory concepts.
Explain the connection between chance regularities and statistical models.
Explain the notion of statistical adequacy and discuss its importance for statistical inference.
Under what circumstances can the modeler claim that the observed data constitute unprejudiced evidence in assessing the empirical adequacy of a theory?
“Statistical inference is a hybrid of a deductive and an inductive procedure.” Explain and discuss.
Discuss the claim: “If the sample size is not large enough for validating the model assumptions, then it is not large enough for reliable inference.”
(a) Explain the main differences between descriptive and inferential (proper) statistics as they relate to their objectives and the role of probability.(b) “There is no such thing as descriptive
(a) Compare and contrast Figures 2.1 and 2.2 in terms of the type of chance regularity pattern they exhibit.(b) Explain intuitively why the descriptive statistics for the x0 data (Figure 2.1) based
InExample2.11 on casting three dice and adding up the dots, explain the different permutations for the occurrence of the events (11, 12) and evaluate their probabilities using Galileo’s reasoning.
(a) Explain the difference between combinations and permutations.(b) Compare and contrast the following sample survey procedures: (i) simple random sampling; (ii) stratified sampling; (iii) cluster
(a) Which of the following observable phenomena can be considered as random experiments, as defined by conditions [a]–[c]. Explain your answer briefly.(i) A die is cast and the number of dots
ForthesetsA = {2, 4, 6} and B = {4, 8, 12}, derive the following:(a) A ∪ B, (b) A ∩ B, (c) A ∪ B relative to S = {2, 4, 6, 8, 10, 12}.Illustrate your answers using Venn diagrams.
A die is cast and the number of dots facing up is counted.(a) Specify the set of all possible outcomes.(b) Define the sets A – the outcome is an odd number, B – the outcome is an even number, C
(a) “Two mutually exclusive events A and B cannot be independent.” Discuss.(b) Explain the notions of mutually exclusive events and a partition of an outcomes set S. How is the latter useful in
Define the concept of a σ-field and explain why we need such a concept for the set of all events of interest. Explain why we cannot use the power set as the event space in all cases.
Consider the outcomes set S = {2, 4, 6, 8} and let A = {2, 4} and B = {4, 6} be the events of interest. Show that the field generated by these two events coincides with the power set of S.
Explain how intervals of the form (−∞, x] can be used to define intervals such as{a}, (a, b), [a, b), (a, b], [a,∞), using set-theoretic operations.
(a) Explain the difference between a relation and a function.(b) Let the domain and co-domain of a numerical relation be A = {1, 2, 3, 4} and B = {2, 3, 5, 7, 11, 13}, respectively. Explain whether
Explain whether the probability functions defined below are proper ones: (i) P(A) = P(A) = }, P(S) = 1, P() = 0, (ii) P(A) = }, P(A) = }, P(S) = 1, P(0) = 0, (iii) P(A) = P(A) = , P(S) = 0, P(0)= 1,
(a) Explain how we can define a simple probability distribution in the case where the outcomes set is finite.(b) Explain how we can define the probability of an event A in the case where the outcomes
Describe briefly, using your own words, the formalization of conditions [a] and [b] of a random experiment into a probability space (S, , P(.)).
Explain how Theorem 2.4: for events A and B: P(B) = P(A ∩ B)+P(A ∩ B) relates to the total probability formula: P(B) = P(A)·P(B|A) + P(A)·P(B|A).
Draw a ball from an urn containing 6 black, 8 white, and 10 yellow balls.(a) Evaluate the probabilities of drawing a black (B), a white (W), or a yellow (Y)ball, separately.(b) Evaluate the
Apply the reasoning in Example 2.50 using the probabilities P(A) = .3, P(B|A) = .95, P(B|A) = .05 and contrast your results with those in that example.
In the Monty Hall puzzle (Example 2.51), explain why the reasoning used by the math professor to reach the conclusion that keeping the original door or switching to the other will make no difference
Describe briefly the formalization of condition [c] of a random experiment into a simple sampling space GIID n .
Explain the notions of independent events and identically distributed trials.
Explain how conditioning can be used to define independence; give examples.
Explain the difference between a sampling space in general and the simple sampling space GIID n in particular.
In the context of the random experiment of tossing a coin twice, derive the probability of event A = {(HT), (TH)} given event B = {(HH), (HT)}. Explain why events A and B are independent.25*. For two
Explain why the abstract probability space is inappropriate for modeling purposes.
(a) “Arandom variable is neither random nor a variable.” Discuss.(b) “The concept of a random variable is a relative concept.” Discuss.(c) Explain the difference between the inverse and the
Consider the random experiment of casting two dice and counting the total number of dots appearing on the uppermost faces. The random variable X takes the value 0 when the total number of dots is odd
Discuss the difference between the following probability set functions in terms of their domain:P(X ≤ x) = PX−1((−∞, x]) = P((−∞, x]).
In the case of the random experiment of “tossing a coin twice”:(a) Which of the functions (i)–(iii) constitute random variables with respect to ?(b) Compare the σ-fields generated by each
Compare and contrast the concepts of a discrete random variable and a continuous random variable.
Describe briefly the transformation of the probability space (S, , P(.)) into a probability model of the form = {f (x; θ), θ ∈ , x ∈ RX}.Explain the relationship between the components of
Explain the main components of a generic probability model above.
Why do we care about the moments of a distribution? How do the moments provide a way to interpret the unknown parameters?
For the exponential distribution, the density function is f (x; θ) = θe−θx, θ>0, x>0.(a) Derive its mean and variance.(b) Derive its mode.
Consider the function f (x) = 140 x3(1 − x)3, 0
Consider the function f (x) = x/2, 0
Consider the discrete random variable X whose distribution is given below:(a) Derive its mean, variance, skewness, and kurtosis coefficients.(b) Derive its mode and coefficient of variation. X -1 0 1
(a) State the properties of a density function.(b) Contrast the properties of the expected value and variance operators.(c) Let X1 and X2 be two independent random variables with the same mean μ and
Explain how the properties of the variance are actually determined by those of the mean operator.
Explain how the moment generating function can be used to derive the moments.
Explain the concept of skewness and discuss why α3 = 0 does not imply that the distribution in question is symmetric.
Explain the concept of kurtosis and discuss why it is of limited value when the distribution is non-symmetric.
For a Weibull distribution with parameters (α = 3.345, β = 3.45), derive the kurtosis coefficient using the formulae in Appendix 3.A.
Explain why matching moments between two distributions can lead to misleading conclusions.
Compare and contrast the cumulative distribution function and the quantile function.
Explain the concepts of a percentile and a quantile and how they are related.
Why do we care about probabilistic inequalities?
“Moments do not characterize distributions in general and when they do we often need an infinite number of moments for the characterization.” Discuss.
Explain the probability integral and the probability integral transformations. How useful can they be in simulating non-uniform random variables?
Consider the discrete uniform distribution with density fx(x; θ) = 1/(n + 1), where n is an integer, x = 0,1,2,...,n. Derive E(X) and Var(X); note that ok (n+1)/2, ok = n (2n+1)(n+1)/6. =
“Marginalizing amounts to throwing away all the information relating to the random variable we are summing (integrating) out.” Comment.
Consider the random experiment of tossing a coin twice and define the random variables X=number of heads, and Y=|number of heads − number of tails|. Derive the joint distribution of (X, Y),
Let the joint density function of two random variables X and Y be(a) Derive the marginal distributions of X and Y.(b) Determine whether X and Y are independent.(c) Verify your answer in (b) using the
Define the concept of independence for two random variables X and Y in terms of the joint, marginal, and conditional density functions.
Explain the concept of a random sample and explain why it is often restrictive for most economic data series.
Describe briefly the formalization of the condition [c] the experiment can be repeated under identical conditions in the form of the concept of a random sample.
Explain intuitively why it makes sense that when the joint distribution f (x, y) is Normal, the marginal distributions fx(x) and fy(y) are also Normal.
Define the raw and central joint moments and show that Cov(X, Y) = E(XY) −E(X)·E(Y). Why do we care about these moments?
(a) Explain the concept of an ordered sample.(b) Explain intuitively why an ordered random sample is neither independent nor identically distributed.
Explain the concepts of identifiability and parameterization.
“In relating statistical models to (economic) theoretical models we often need to reparameterize/restrict the former in order to render the estimated parameters theoretically meaningful.” Explain.
Explain the concept of a random sample and its restrictiveness in the case of most economic data series.
How do we assess the distributional features of a data series using a t-plot?
“A smoothed histogram is more appropriate in assessing the distributional features of a data series than the histogram itself because the former is less data specific.” Explain.
Explain how one can distinguish between a t-plot of NIID and a t-plot of Student’s t IID observations.
Explain the relationship between the abstract concept of independence and the corresponding chance regularity pattern in a t-plot of a data series.
Explain how any form of dependence will help the modeler in prediction.
Explain the relationship between the abstract concept of identical distribution and the corresponding chance regularity pattern in a t-plot of a data series.
“Without an ordering of the observations one cannot talk about dependence and heterogeneity.” Discuss.
Explain the notion of a P–P plot and the Normal P–P plot in particular.
Compare and contrast a Normal P–P and a Normal Q–Q plot.
Explain how the standardized Student’s t P–P plot can be used to evaluate the degrees of freedom parameter.
Explain the notion of a reference distribution in a P–P plot. Why does the Cauchy reference distribution take different shapes in the context of a standardized Normal and a Student’s t P–P plot?
The data in Appendix 5.A (see Lai and Xing (2008), p. 71) denote logreturns on six stocks, monthly observations from August 2000 to October 2005, where PFE=Pfizer, INTEL=Intel, CITI=Citigroup,
Why do we care about heterogeneity and dependence in statistical models?
Explain how the idea of sequential conditioning helps to deal with the problem of many dimensions of the joint distribution of a non-random sample.
(a) Define the following concepts: (i) joint moments, (ii) conditional moments,(iii) non-correlation, (iv) orthogonality.(b) Explain the difference between (i) dependence vs. correlation and (ii)
ForXN(0, 1), define the random variable Y = X2 − 1 and show that Cov(X, Y) = 0, but the two random variables are not independent.
Let the joint density function of two random variables X and Y be(a) Derive the conditional distributions f (y|x), x = 0, 1.(b) Derive the following moments: E(X), E(Y), Var(X), Var(Y), Cov(X, Y),
Explain the notion of rth-order conditional dependence and compare it with that of(m, k)th-order dependence.
Explain and compare conditional independence and Markov dependence.
Explain why non-correlation implies independence in the case of a bivariate Normal distribution. How does one assess the correlation by looking at a scatterplot of observed data?

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