Classical and Axiomatic Approaches in Probability Theory
30 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What does the Law of Total Probability express?

  • The total probability of an event in terms of conditional probabilities (correct)
  • The probability of the joint occurrence of two or more events
  • The probability of mutually exclusive events
  • The compound probability of independent events

How is compound probability calculated for independent events?

  • $P(A|B) \times P(B)$
  • $P(A) + P(B)$
  • $P(A) \times P(B)$ (correct)
  • $P(B|A) \times P(A)$

What does conditional probability measure?

  • The total probability of an event in terms of conditional probabilities
  • The compound probability of dependent events
  • The joint occurrence of two or more events
  • The likelihood of an event occurring given that another event has already occurred (correct)

What does Bayes' theorem provide a way to update?

<p>The posterior probability of a hypothesis based on new evidence (B)</p> Signup and view all the answers

How is compound probability calculated for dependent events?

<p>$P(A|B) \times P(B)$ (D)</p> Signup and view all the answers

What is the formula for the Law of Total Probability?

<p>$P(A) = \sum_{i=1}^{n} P(A|B_i) \times P(B_i)$ (B)</p> Signup and view all the answers

What is the main concept described by the Central Limit Theorem?

<p>Distribution of sample means (D)</p> Signup and view all the answers

In the context of random variables, what does 'almost sure convergence' refer to?

<p>Convergence with probability 1 (A)</p> Signup and view all the answers

Which property must the joint probability mass function satisfy for two discrete random variables X and Y?

<p>Non-negativity for all values of X and Y (A)</p> Signup and view all the answers

What does the Central Limit Theorem state about the distribution of sample means as the sample size increases?

<p>It approximates a normal distribution (B)</p> Signup and view all the answers

How are marginal distributions derived from the joint distribution of two random variables?

<p>By integrating over all possible values of one variable (C)</p> Signup and view all the answers

What do covariance and correlation describe in relation to two random variables?

<p>Relationship and dependency between variables (A)</p> Signup and view all the answers

What type of random variable can take on any value within a given range?

<p>Continuous random variable (A)</p> Signup and view all the answers

Which function specifies the relative likelihood of a continuous random variable taking on a particular value?

<p>Probability density function (B)</p> Signup and view all the answers

How is the expected value of a continuous random variable calculated?

<p>By integrating over the PDF (B)</p> Signup and view all the answers

Which function is used to analyze the distribution of random variables and calculate moments?

<p>Generating function (D)</p> Signup and view all the answers

In the context of random variables, what does the moment generating function (MGF) provide a convenient way to extract information about?

<p>Distribution of random variables (D)</p> Signup and view all the answers

Which law of large numbers describes convergence almost surely as the number of observations increases to infinity?

<p>Strong Law of Large Numbers (SLLN) (B)</p> Signup and view all the answers

What is the main difference between the classical and axiomatic approaches to defining probability?

<p>The classical approach assumes all outcomes are equally likely, while the axiomatic approach is based on fundamental principles. (C)</p> Signup and view all the answers

Which mathematician introduced the axiomatic approach to defining probability in the 20th century?

<p>Andrey Kolmogorov (D)</p> Signup and view all the answers

In the classical approach, what is the formula used to calculate the probability of an event E occurring?

<p>$P(E) = \frac{n(E)}{n(S)}$ (C)</p> Signup and view all the answers

Which axiom states that the probability of any event is a non-negative real number?

<p>Non-negativity (B)</p> Signup and view all the answers

What is the main principle behind the axiomatic approach to defining probability?

<p>Fundamental principles or axioms (C)</p> Signup and view all the answers

Which approach has become the standard framework for defining probability in modern probability theory?

<p>Axiomatic Approach (C)</p> Signup and view all the answers

What is the formula to calculate the marginal PMF of a discrete random variable X?

<p>P(X=x) = ∑y P(X=x, Y=y) (D)</p> Signup and view all the answers

How is the conditional PMF of a discrete random variable X given Y=y calculated?

<p>P(X=x | Y=y) = P(X=x, Y=y) / P(Y=y) (A)</p> Signup and view all the answers

What is the formula to calculate the marginal PDF of a continuous random variable X?

<p>fX?(x) = ∫ fX,Y?(x,y) dy (D)</p> Signup and view all the answers

How is the conditional PDF of a continuous random variable X given Y=y calculated?

<p>$f_{X|Y}(x|y) = \frac{f_{Y}(y)}{f_{X,Y}(x,y)}$ (D)</p> Signup and view all the answers

What does the conditional distribution of one random variable given another represent?

<p>The distribution of the first variable when the second variable is fixed at a specific value (D)</p> Signup and view all the answers

In the context of conditional distributions, what is the role of the marginal PMF/PDF?

<p>Forming a bridge between joint and conditional distributions (B)</p> Signup and view all the answers

More Like This

Use Quizgecko on...
Browser
Open
Browser
Browser