Classical and Axiomatic Approaches in Probability Theory
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Questions and Answers

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

  • The classical approach is non-normative, while the axiomatic approach is normative.
  • The classical approach is based on equally likely outcomes, while the axiomatic approach is based on fundamental principles. (correct)
  • The classical approach defines probability through a set of axioms, while the axiomatic approach relies on total number of possible outcomes.
  • The classical approach was introduced by Andrey Kolmogorov, while the axiomatic approach is a more recent development.

    • Which mathematician played a key role in introducing the axiomatic approach to defining probability?

  • Carl Friedrich Gauss
  • Andrey Kolmogorov (correct)
  • Blaise Pascal
  • Leonhard Euler

    • According to the classical approach, how is the probability of an event calculated?

  • $P(E) = n(E) + n(S)$
  • $P(E) = n(E) - n(S)$
  • $P(E) = n(S) - n(E)$
  • $P(E) = \frac{n(E)}{n(S)}$ (correct)

    • Which of the following is one of the basic axioms in the axiomatic approach to defining probability?

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

    In the classical approach, what does it mean when outcomes are assumed to be equally likely?

    <p>Each outcome in the sample space has an equal chance of occurring.</p> Signup and view all the answers

    What does the normalization axiom state in the axiomatic approach to defining probability?

    <p>The probability of the entire sample space is 1.</p> Signup and view all the answers

    What does the probability mass function (PMF) provide for a discrete random variable?

    <p>The probability of each possible value</p> Signup and view all the answers

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

    <p>Sum of each possible value weighted by its probability</p> Signup and view all the answers

    Which function helps analyze the distribution of discrete random variables?

    <p>Probability generating function (PGF)</p> Signup and view all the answers

    What type of convergence is associated with the weak law of large numbers?

    <p>Convergence in probability</p> Signup and view all the answers

    What does the strong law of large numbers state about sample averages?

    <p>Converges almost surely to expected value</p> Signup and view all the answers

    What does the Law of Total Probability state?

    <p>The total probability of an event can be expressed as the sum of the probabilities of A given each partition Bi.</p> Signup and view all the answers

    What is compound probability?

    <p>The probability of the intersection of two or more events.</p> Signup and view all the answers

    How is compound probability calculated for independent events A and B?

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

    What is conditional probability?

    <p>The likelihood of an event occurring given that another event has already occurred.</p> Signup and view all the answers

    In Bayes' theorem, what does P(B|A) represent?

    <p>The likelihood of observing evidence B given that hypothesis A is true.</p> Signup and view all the answers

    How can compound probability be calculated for dependent events A and B?

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

    According to the Central Limit Theorem, as the sample size increases, what distribution does the sample mean approach?

    <p>Normal distribution</p> Signup and view all the answers

    What is the main application of the Central Limit Theorem in statistics?

    <p>Construction of confidence intervals</p> Signup and view all the answers

    In the context of joint distributions, what does the marginal distribution describe?

    <p>Probability distribution of one variable without considering others</p> Signup and view all the answers

    Which property must the joint probability mass function satisfy for discrete random variables?

    <p>$fX,Y(x,y) &gt; 0$ for all $x$ and $y$</p> Signup and view all the answers

    What concept describes the relationship between two random variables in terms of how they change together?

    <p>Covariance</p> Signup and view all the answers

    How are conditional distributions derived from joint distributions?

    <p>By dividing the joint distribution by the marginal distribution</p> Signup and view all the answers

    What is the formula to calculate the marginal PMF of X for discrete random variables?

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

    How is the conditional PMF of X given Y=y obtained for continuous random variables?

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

    What does the conditional distribution of one random variable given the value of another random variable 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

    For discrete random variables, how is the conditional PMF obtained?

    <p>By dividing the joint PMF by the marginal PMF of the conditioned variable</p> Signup and view all the answers

    What is the formula to calculate the marginal PDF of X for continuous random variables?

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

    How can the conditional PDF of X given Y=y be described for continuous random variables?

    <p>$fX?Y?(x?y) = fY?(y)/fX,Y?(x,y)$</p> Signup and view all the answers

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