Probability Generating Functions and Ball Transfer
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Questions and Answers

What is the variance of the random variable W based on the given generating function GW(s) = s3?

  • 0
  • 1 (correct)
  • 3
  • Undefined

    • Which equation represents the variance of the random variable Y in GY(s) = (1/3) / (1 – s/2)?

  • 4
  • 2/3 (correct)
  • 1
  • 3

    • If we consider the toss of a biased coin until observing r heads, what is the variance for the random variable X after tossing the fair coin Y times?

  • rp(1 - p)
  • r(1 - p)/p
  • r(1 - p)/p2 (correct)
  • r(1 + p)/p2

    • What is the expected value E[X] when r = 1 for the scenario described with the biased coin?

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

    Considering the binomial distribution, what is the correct formula for variance given n trials and probability p?

    <p>np(1 - p)</p> Signup and view all the answers

    What is the marginal probability density function (pdf) of Y, given that Y > 2?

    <p>$\frac{1}{y^2}$</p> Signup and view all the answers

    What expression represents the conditional pdf of X given Y = y?

    <p>$12x^2y^3e^{-xy}$</p> Signup and view all the answers

    How is the expected value of X given Y = y calculated?

    <p>By integrating $xf(x|y)$ over x</p> Signup and view all the answers

    What is the expected value of X given Y as a function of Y?

    <p>$\frac{3}{Y}$</p> Signup and view all the answers

    What is the reason E[X|Y] is considered a random variable?

    <p>It depends on the variable Y</p> Signup and view all the answers

    Using double-averaging, what is the resulting value for E[X]?

    <p>$0.75$</p> Signup and view all the answers

    What is the limiting distribution of Y in the coin tossing scenario?

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

    Which of the following integrals corresponds to E[X|Y]?

    <p>$\int_0^\infty x^2y e^{-xy} dx$</p> Signup and view all the answers

    What is the conditional distribution of X given Y = y?

    <p>X | Y = y ~ NB(y, p)</p> Signup and view all the answers

    What is the formula for calculating P(X < 3)?

    <p>P(X &lt; 3) = P(X = 1) + P(X = 2)</p> Signup and view all the answers

    What is the expected value E[X] when finding E[Y]/p?

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

    What is the equation used to find the variance Var(X)?

    <p>Var(X) = E[Var(X|Y)] + Var(E[X|Y])</p> Signup and view all the answers

    What does the waiting time until the first occurrence of 'ABRACADABRA' represent?

    <p>Delayed renewal event</p> Signup and view all the answers

    How is the expected number of trials until the first occurrence of 'ABRA' determined?

    <p>Using the Renewal Theorem</p> Signup and view all the answers

    What is the probability generating function FABRA(s) based on the given sequence?

    <p>FABRA(s) = [s^4/625]/[1 - s + (1 - s)s^3/125 + s^4/625]</p> Signup and view all the answers

    What is the probability that 'ABRA' occurs for the first time on trial 7?

    <p>(1/5)^3</p> Signup and view all the answers

    What is the expected value $E[X]$ representing in Harry Potter's scenario?

    <p>The average time taken to reach the Hall of Prophecies</p> Signup and view all the answers

    Which aspect of the problem does the variance $Var(X)$ measure?

    <p>The unpredictability of Harry's time to the goal</p> Signup and view all the answers

    What would be a method to calculate $E[X]$ if Hermione labels the doors?

    <p>Average all possible outcomes without repeating any doors</p> Signup and view all the answers

    What does it indicate if $E[Vλ] = ∞$ in a renewal process?

    <p>The average time until renewal does not exist</p> Signup and view all the answers

    In the context of a generating function $Fλ(s)$, what does the condition $F(1) = 1$ signify?

    <p>The probabilities sum to one for the events</p> Signup and view all the answers

    Which requirement must be met for $E[Tλ]$ to show positive recurrence?

    <p>It must be finite</p> Signup and view all the answers

    What does the joint probability density function for continuous random variables X and Y describe?

    <p>The likelihood of various combinations of values for two variables</p> Signup and view all the answers

    In the context of Harry's problem, what is the significance of the time delays (2 minutes, 4 minutes, 10 minutes) associated with each door?

    <p>They influence the average time to the goal</p> Signup and view all the answers

    What is the marginal probability density function (pdf) of Y, denoted as f(y)?

    <p>f(y) = (1/(α − 1)!) * y^(α−1)</p> Signup and view all the answers

    What is the expected value of X given Y = y, denoted as E[X|Y = y]?

    <p>E[X|Y = y] = y * 173</p> Signup and view all the answers

    What is the method to find the expected value of X, denoted as E[X], using double averaging?

    <p>E[X] = ∫E[X|Y] f(y) dy</p> Signup and view all the answers

    What is the probability that you get the highest mark (i.e., P(X=173)) when marks are independent?

    <p>P(X=173) = (1/174)</p> Signup and view all the answers

    When determining the probability that a transferred ball is white, which approach is most appropriate?

    <p>Use the total probability theorem considering all outcomes.</p> Signup and view all the answers

    Given that a ball was transferred and you select a ball from the box, what is the probability that the selected ball is white?

    <p>The probability is 0.579 due to the four cases considered.</p> Signup and view all the answers

    What defines a random variable as 'null proper'?

    <p>Its expected value is equal to zero.</p> Signup and view all the answers

    In order to classify a probability generating function as 'short proper', what condition must be checked?

    <p>There must exist a finite variance.</p> Signup and view all the answers

    Study Notes

    Probability Generating Functions

    • If a probability generating function (pgf) has a finite radius of convergence, a value of 1 at that radius, and a finite derivative at that value, the random variable is short proper.
    • If the pgf has a radius of convergence of 1 with a finite derivative at 1 and a finite second derivative at 1, the random variable is null proper.
    • If the pgf does not meet the above criteria, the random variable is improper.
    • To determine the variance, calculate the second derivative of the pgf and evaluate at 1.

    Probability of a White Ball in a Box

    • There is a 0.5 probability of choosing either box initially.
    • If you initially choose the box with 2 white and 1 black balls, then the probability of transferring a white ball is 2/3.
    • If you initially choose the box with 2 white and 2 black balls, then the probability of transferring a white ball is 1/2.
    • Therefore, the overall probability of transferring a white ball is (0.5 * 2/3) + (0.5 * 1/2) = 0.579.
    • Given that a white ball is selected from the box with the transferred ball, the probability that a white ball was transferred is 0.8.

    Expected Number of Trials

    • The expected number of trials to observe a sequence, like "ABRACADABRA", is determined by the renewal theorem, which states that the expected waiting time is equal to the sum of the expected waiting times for the overlapping subsequences.
    • In the case of "ABRACADABRA", the maximum overlap subsequence is "ABRA", and the expected waiting time for that subsequence is 5 trials.
    • The overall expected waiting time for "ABRACADABRA" is then 5 (for "TA") + 5 (for "ABRA") + 5 (for "ABRACADABRA") = 48,828,755 trials.

    Harry Potter's Escape

    • The expected time for Harry to reach his goal is 8 minutes calculated by the weighted average of the time it takes to reach the goal via each door.
    • The variance is 26 minutes.
    • If the doors are labeled, Harry only needs to enter each door once, so the expected time is 10 minutes.
    • The variance is 0.

    Marginal and Conditional Probability Density Functions

    • The marginal pdf of Y is the integral of the joint pdf over all possible values of X, which is (3-1)!/y^2 for y > 2.
    • The conditional pdf of X given Y = y is found by dividing the joint pdf by the marginal pdf, which is 12x^2 y^3e^(-xy) for x > 0.
    • The expected value of X given Y = y is found by integrating the product of X and the conditional pdf over all possible values of X, which is 6/(2y).
    • The expected value of X given Y is E[X|Y] = 3/Y, a random variable because it is a function of Y.
    • The expected value of X can be found using double averaging, which is the expected value of the expected value of X given Y, resulting in E[X] = 6/2^3 = 0.75.

    Probability of Getting the Highest Mark

    • The probability of getting the highest mark is 1/(174) because all marks are equally likely.
    • The expected number of students with lower marks than you, given your mark is y, is 173*y.
    • The expected number of students with lower marks than you is the expected value of the expected number of students with lower marks given your mark, which is 173/2.

    Conditional Variance

    • The conditional variance is the variance of a random variable conditioned on another random variable, which can be calculated using the formula Var(X) = E[Var(X|Y)] + Var(E[X|Y]).
    • In this situation, the conditional variance of X is E[Y(1 – p)/p^2] + Var(Y/p), which simplifies to (4 – 2p)/p.

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    Description

    This quiz covers concepts related to probability generating functions, including classifications of random variables as short proper, null proper, or improper based on their properties. Additionally, it explores the probability of transferring a white ball from two different boxes containing white and black balls. Test your knowledge on these essential probability topics!

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