Discrete and Continuous Random Vectors
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Questions and Answers

What is the function pX in the context of a discrete random vector X?

  • The expectation of X
  • The joint probability density function of X
  • The cumulative distribution function of X
  • The probability mass function of X (correct)

    • Which definition of probability does P2 satisfy?

  • Frequentist definition
  • Laplace definition
  • Kolmogorov axiomatic definition (correct)
  • None of the above

    • What is the notation P(X = x) equivalent to in the context of a discrete random vector X?

  • PX(x)
  • FX(x)
  • pX(x) (correct)
  • fX(x)

    • What is the condition for a function fX to be a joint probability density function of a continuous random vector X?

    <p>fX is non-negative and integrates to 1</p> Signup and view all the answers

    What is the probability of the outcome {H} in P2?

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

    What is the relationship between the joint probability density function fX and the cumulative distribution function FX of a continuous random vector X?

    <p>FX(x) = ∂/∂x1 ... ∂/∂xp fX(x) for almost all x ∈ Rp</p> Signup and view all the answers

    What is the probability of the empty set in P2?

    <p>0</p> Signup and view all the answers

    How is the probability of a subset A with more than one element defined in Example 1.5?

    <p>As the sum of probabilities of individual outcomes</p> Signup and view all the answers

    What is the property of the joint probability density function fX of a continuous random vector X?

    <p>fX is non-negative</p> Signup and view all the answers

    What is the joint probability density function of a uniform random vector X ∼ U([0, 1] × [0, 2])?

    <p>f(x1, x2) = 1/2, (x1, x2)′ ∈ [0, 1] × [0, 2]</p> Signup and view all the answers

    What is the probability of the outcome {0} in Example 1.5?

    <p>0.018</p> Signup and view all the answers

    What is the purpose of the definition of probability in Example 1.5?

    <p>To free the mathematical meaning of probability from the random experiment</p> Signup and view all the answers

    What is the value of P1X ((0, 1])?

    <p>1/2</p> Signup and view all the answers

    What is the cumulative distribution function of a random variable X?

    <p>FX (x) = PX ((−∞, x])</p> Signup and view all the answers

    What is the induced probability of a random variable X1?

    <p>PX1 (B) = P1 (N0 ∩ B) p</p> Signup and view all the answers

    What is the main distinction in types of random variables?

    <p>Discrete and continuous</p> Signup and view all the answers

    What is the purpose of specifying a probability model?

    <p>To introduce a shortcut for computing probabilities</p> Signup and view all the answers

    What does PX ((−∞, x]) denote?

    <p>The probability that X is less than or equal to x</p> Signup and view all the answers

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