AP Statistics: Probability rules and distributions

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Questions and Answers

Consider a scenario where a novel stochastic process exhibits asymptotic behavior such that the long-run relative frequency of an event $E$ converges to a value dictated by a transcendental equation involving the Riemann zeta function. Which of the following statements best describes the limitations of applying the Law of Large Numbers in this specific context?

The Law of Large Numbers guarantees convergence to a frequentist probability, but provides no insight into the sample size required to reach a specific level of accuracy when transcendental functions are involved in defining probabilities.
The Law of Large Numbers is inapplicable because the transcendental nature of the limit violates the axioms of probability theory.
While the Law of Large Numbers still holds, obtaining an accurate estimate within a reasonable computational time becomes exceptionally challenging due to the slow rate of convergence dictated by the properties of the Riemann zeta function. (correct)
The Law of Large Numbers is only valid for discrete random variables, and the Riemann zeta function implies a continuous probability distribution.

Given two events, A and B, within a sample space $\Omega$, the assertion that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ always provides an accurate computation of the probability of either A or B occurring, assuming all probabilities are well-defined and measurable according to Kolmogorov's axioms.

True (A)

In the context of Bayesian epistemology, where prior probabilities are updated based on observed evidence, what is the philosophical significance of assigning a non-zero prior probability to every conceivable hypothesis, even those considered highly implausible?

Prevents absolute certainty and allows for continuous refinement of beliefs.

In quantum probability theory, which deviates from classical probability by allowing for non-commutative events, the violation of ______'s inequality serves as a key indicator of quantum entanglement and non-classical correlations. If the inequality is not violated, the system can usually be described by classical physical systems.

Bell Signup and view all the answers

Match the following probability concepts with their applications in advanced statistical modeling:

Bayesian Inference = Updating prior beliefs with evidence through Bayes' Theorem for parameter estimation and model comparison. Markov Chain Monte Carlo (MCMC) = Employing Markov chains to sample from complex probability distributions, crucial in Bayesian computation and simulation. Dirichlet Process = A stochastic process used as a prior in Bayesian nonparametrics, allowing for flexible modeling of unknown distributions. Copula Functions = Modeling the dependence structure between random variables independently from their marginal distributions. Signup and view all the answers

Consider a complex system where the probability of event A occurring depends on a continuous-time Markov process $X(t)$ governing environmental conditions, such that $P(A|X(t)) = e^{-\lambda X(t)}$, where $\lambda$ is a positive constant. Given that $X(t)$ follows an Ornstein-Uhlenbeck process with mean reversion level $\mu$ and volatility $\sigma$, what is the most appropriate method for estimating the long-term average probability of event A?

Using a Monte Carlo simulation to approximate the time average of $P(A|X(t))$ over a long time horizon, effectively calculating $\frac{1}{T} \int_0^T e^{-\lambda X(t)} dt$. (C) Signup and view all the answers

If two events, A and B, are independent according to one probability measure P, they must necessarily also be independent under any other probability measure Q defined on the same sample space, provided that both P and Q assign non-zero probabilities to both A and B.

False (B) Signup and view all the answers

In the context of algorithmic fairness, explain how disparate impact can arise even when a classification algorithm is explicitly designed to be 'color-blind' (i.e., not directly using sensitive attributes), and suggest a method to mitigate this issue.

Proxy variables correlate with protected attributes. Use a causal inference framework to build a model. Signup and view all the answers

The ______ paradox demonstrates that aggregating conditional probabilities across different subgroups can yield results that contradict the marginal probabilities, highlighting the importance of considering confounding variables when interpreting statistical data.

Simpson Signup and view all the answers

Consider a scenario where a clinical trial is designed to evaluate the efficacy of a new drug. The probability of a patient recovering with the drug is denoted as $P(R|D)$, and without the drug, the probability is $P(R|¬D)$. However, there exists an unobserved confounder $C$ (e.g., genetic predisposition) that affects both the likelihood of receiving the drug and the probability of recovery. Which of the following statistical techniques is most appropriate for estimating the causal effect of the drug on recovery, accounting for the unobserved confounder?

Instrumental variables (IV) regression using a valid instrument that affects drug assignment but is independent of recovery except through its effect on drug assignment. (D) Signup and view all the answers

Flashcards

Random Process

A process where outcomes are uncertain and determined by chance.

Long Run Relative Frequency

The viewpoint that probability is the long-run relative frequency of an outcome.