Classical and Axiomatic Approaches in Probability Theory
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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</p> Signup and view all the answers

    How is compound probability calculated for dependent events?

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

    What is the formula for the Law of Total Probability?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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