Algorithm for Random Number Generation and Dice
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Questions and Answers

What is the smallest value of k for which $a^k \equiv 1 \mod m$ when choosing a suitable parameter for generating random numbers?

  • m - 2
  • m - 1 (correct)
  • m + 1
  • m

    • The probability for each face of a fair six-sided die is equal.

    True

    What is the empirical mean for the results of rolling a six-sided die 10,000 times, based on the given data?

    3.6192

    In the context of generating random numbers, the output after performing the operation is denoted as $x_k = a imes x_{k-1} \mod m$, and $x_k$ represents the ______.

    <p>pseudo-random number</p> Signup and view all the answers

    Match each term with its correct description:

    <p>Random Number Generator = A process for generating a sequence of numbers that do not have any predictable pattern Empirical Variance = A measure of how much the numbers in a sample deviate from the mean Sample Space = The set of all possible outcomes of a random experiment Histogram = A graphical representation of the distribution of numerical data</p> Signup and view all the answers

    What kind of distribution is expected when rolling a fair die multiple times?

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

    When generating pseudo-random numbers, the period will always be the same no matter the choice of parameters.

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

    What command in Python is used to generate a random integer between 1 and 6?

    <p>random.randint(1, 6)</p> Signup and view all the answers

    What is the probability of rolling an odd number when rolling a fair six-sided die?

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

    In a discrete probability space, the sum of the probabilities of all possible outcomes equals 1.

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

    If the sample space is Ω = {1, 2, 3, 4}, what is the probability function p(ω) if it is uniformly distributed?

    <p>1/4</p> Signup and view all the answers

    The event of rolling a number greater than 4 is represented by the set ______.

    <p>{5, 6}</p> Signup and view all the answers

    Match the following events with their descriptions:

    <p>A∩B = The intersection of sets A and B A∪B = The union of sets A and B A extbackslash B = The difference of sets A and B Ω extbackslash A = The complement of set A</p> Signup and view all the answers

    What is the correct formula for the probability of an event A in a discrete probability space?

    <p>P(A) = Σ p(ω) where ω ∈ A</p> Signup and view all the answers

    If A and B are two events such that A∩B = ∅, they are considered independent events.

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

    What does it mean for p(ω) to be defined as p(ω) = 1/#Ω in a uniform distribution?

    <p>Each outcome has an equal probability.</p> Signup and view all the answers

    What is the formula for calculating the unbiased empirical variance?

    <p>$ rac{1}{N-1} imes ext{sum}(x_i - ar{x})^2$</p> Signup and view all the answers

    In a discrete probability space, the sum of probabilities of all outcomes must equal 1.

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

    What are the three axioms of probability defined by Kolmogorov?

    <ol> <li>Probability lies between 0 and 1; 2. The sure event has probability 1; 3. Probability of disjoint events' union equals the sum of their probabilities.</li> </ol> Signup and view all the answers

    A probability function maps events to [0, 1]. This can be written as __________.

    <p>p : P(Ω) → R</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Sample Space = Set of all possible outcomes Event = Subset of the sample space Outcome = An element of the sample space Probability Function = Maps events to probabilities</p> Signup and view all the answers

    If the sample space is $ ext{Ω} = ext{Set of natural numbers}$, what does $ ext{P(Ω)}$ represent?

    <p>The power set of Ω</p> Signup and view all the answers

    An event can have a probability greater than 1.

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

    What is the median of the data set {2, 4, 6, 8}?

    <p>5</p> Signup and view all the answers

    What is the condition for using the Poisson approximation according to the theorem?

    <p>lim n→∞ n · qn must equal λ</p> Signup and view all the answers

    For any A ⊆ Ω, A ∩ B ⊆ B holds true.

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

    What does the conditional distribution P(A | B) represent?

    <p>The probability of event A occurring given that event B has occurred.</p> Signup and view all the answers

    The _____ of total probability helps in calculating probabilities when there are multiple disjoint events.

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

    Match the following definitions with their corresponding terms:

    <p>Conditional Distribution = P(A | B) Disjoint Events = Events that cannot occur simultaneously Discrete Probability Measure = A function satisfying the Kolmogorov axioms Poisson Distribution = A discrete distribution with parameter λ</p> Signup and view all the answers

    According to Proposition 3.2, what is true about P( · | B)?

    <p>It is a discrete probability measure.</p> Signup and view all the answers

    The statement P(Ω | B) = P(B) is equal to 1.

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

    What is the requirement for P(B) in the definition of conditional distribution?

    <p>P(B) must be greater than 0.</p> Signup and view all the answers

    What is the probability that an individual is sick given a positive test result based on the calculations provided?

    <p>0.0902</p> Signup and view all the answers

    Events A and B are independent if and only if P(A ∩ B) is equal to P(A) + P(B).

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

    If P(A|B) is 0.99 and P(B) is 0.001, what is the value of P(B|A) in the provided equation?

    <p>0.0902</p> Signup and view all the answers

    The probability of multiple events occurring can be expressed as P(A1 ∩ A2) = P(A1) · P(A2 | A1). Fill in the blank: P(A1 ∩ A2 ∩ A3) = P(A1) · P(A2 | A1) · P(A3 | A1 ∩ ____.

    <p>A2</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Theorem 3.4 = Application of conditional probabilities Lemma 3.6 = Product of probabilities for sequential events Independent Events = Events whose probability remains unchanged Conditional Probability = Probability of an event given another event occurred</p> Signup and view all the answers

    In the discrete probability space (Ω, P), which of the following is a property of independent events?

    <p>P(A ∩ B) = P(A) · P(B)</p> Signup and view all the answers

    The total probability theorem allows for the calculation of P(B|A) using P(A|B) and P(B).

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

    The formula for conditional probability is P(A|B) = P(A ∩ B) / P(___).

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

    Study Notes

    Algorithm 1 for Generating Random Numbers

    • The algorithm generates a sequence of pseudorandom numbers using modulo arithmetic.

    • Input:

      • x0 (initial value, between 0 and 1)
      • m (large prime number)
      • a (natural number)

    • Output:

      • Sequence of numbers x1, x2, ... , xn

    • Operation:

      • The algorithm iterates through k from 1 to n, where n is desired number of random numbers, and for each k, it calculates: xk = a*xk-1 mod m;

    • Choosing a:

      • To ensure good randomness, a must be chosen such that the smallest k for which ak ≡ 1 mod m is k = m - 1.
      • With correct a, the algorithm generates n = m - 1 unique pseudo-random numbers before they start repeating.

    Rolling a Dice

    • Sample space for a six-sided dice:

      • {1, 2, 3, 4, 5, 6}

    • Probability for each face (fair dice):

      • P({1}) = P({2}) = ... = P({6}) = 1/6

    • Python implementation:

      • dice = [random.randint(1, 6) for i in range(10000)]
      • This creates a list of 10000 random numbers between 1 and 6.

    • Mean for a random sample:

      • Formula: x = (1/N) * Σ(xi) where N is the sample size and xi are the individual values of the random sample.

    • Empirical variance:

      • Formula: v̂ = (1/N) * Σ(xi - x)^2, it measures the deviation from the mean
      • Unbiased variance: v = (1/(N-1)) * Σ(xi - x)^2, it is calculated as N-1 in the denominator.

    • Empirical standard deviation:

      • Formula: s = √v and ŝ = √v̂

    • Median for a random sample:

      • Formula: median(x) = (y(n+1)/2) if n is odd, median(x) = (y(n/2)) if n is even.

    Discrete Probability Spaces

    • Sample space (Ω): A set of all possible outcomes of an experiment.
    • Outcomes (ω): Individual elements of the sample space.
    • Events (A): Subsets of the sample space, representing specific combinations of outcomes.
    • Power set (P(Ω)): The set of all possible subsets of the sample space, including the empty set and the sample space itself.

    Probability

    • Probability function (P): A mapping from the power set of the sample space to real numbers between 0 and 1.
    • Kolmogorov's axioms:
      • Axiom 1: For any event A, 0 ≤ P(A) ≤ 1.
      • Axiom 2: P(Ω) = 1.
      • Axiom 3: For any countable set of pairwise disjoint events ( Ai ∩ Aj = ∅) : P(∪Ai) = ΣP(Ai).
    • Discrete probability space (Ω, p): A sample space Ω (at most countable) and a probability function p: Ω→[0, 1] where the sum of probabilities for all outcomes equals 1.

    Conditional Probability and Independence

    • Conditional probability: P(A|B) = P(A ∩ B) / P(B)

      • Represents the probability of event A happening given that event B has already occurred.
      • Requires P(B) > 0.

    • Law of total probability:

      • For a set of events (B1, B2, ..., Bk) that form a partition of the sample space (meaning they are mutually exclusive and cover the whole sample space), then P(A) = P(A ∩ B1) + P(A ∩ B2) + ... + P(A ∩ Bk)

    • Bayes' theorem:

      • P(B | A) = (P(A | B) * P(B)) / P(A), used to calculate the probability of an event given that another event has occurred.

    • Independence of events:

      • Events A and B are independent if: P(A ∩ B) = P(A) * P(B). This means that the occurrence of one event does not affect the probability of the other.

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    Description

    This quiz covers the algorithm for generating pseudorandom numbers using modulo arithmetic and the probability concepts related to rolling a fair six-sided dice. Participants will explore how initial values and parameters affect randomness, as well as basic probability calculations for dice outcomes.

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