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15.6: Renewal Reward Processes
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/15%3A_Renewal_Processes/15.06%3A_Renewal_Reward_Processes
Note first that $R_t = \sum_{i=1}^{N_t} Y_i = \sum_{i=1}^{N_t + 1} Y_i - Y_{N(t) + 1}$ Next Recall that $N_t + 1$ is a stopping time for the sequence of interarrival times $\bs{X}$ for \( t ...Note first that $R_t = \sum_{i=1}^{N_t} Y_i = \sum_{i=1}^{N_t + 1} Y_i - Y_{N(t) + 1}$ Next Recall that $N_t + 1$ is a stopping time for the sequence of interarrival times $\bs{X}$ for $t \in (0, \infty)$ , and hence is also a stopping time for the sequence of interarrival time, reward pairs $\bs{Z}$ . (If a random time is a stopping time for a filtration, then it's a stopping time for any larger filtration.) By Wald's equation, \[ \E\left(\sum_{i=1}^{N_t + 1} Y_i\right) = \nu \…
13.10: Bold Play
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/13%3A_Games_of_Chance/13.10%3A_Bold_Play
This representation is unique except when $x$ is a binary rational (sometimes also called a dyadic rational), that is, a number of the form $k / 2^n$ where $n \in \N_+$ and \(k \in \{1, 3, \ldot...This representation is unique except when $x$ is a binary rational (sometimes also called a dyadic rational), that is, a number of the form $k / 2^n$ where $n \in \N_+$ and $k \in \{1, 3, \ldots, 2^n - 1\}$ ; the positive integer $n$ is called the rank of $x$ . Thus, for $p = \frac{1}{2}$ (fair trials), the probability that the bold gambler reaches the target fortune $a$ starting from the initial fortune $x$ is $x / a$ , just as it is for the timid gambler.
1.2: Functions
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/01%3A_Foundations/1.02%3A_Functions
$x \in f^{-1}(A \cup B)$ if and only if $f(x) \in A \cup B$ if and only if $f(x) \in A$ or $f(x) \in B$ if and only if $x \in f^{-1}(A)$ or $x \in f^{-1}(B)$ if and only if \( x ... $x \in f^{-1}(A \cup B)$ if and only if $f(x) \in A \cup B$ if and only if $f(x) \in A$ or $f(x) \in B$ if and only if $x \in f^{-1}(A)$ or $x \in f^{-1}(B)$ if and only if $x \in f^{-1}(A) \cup f^{-1}(B)$
16.22: Continuous-Time Queuing Chains
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/16%3A_Markov_Processes/16.22%3A_Continuous-Time_Queuing_Chains
When $k = 1$ , the single-server queue, the exponential parameter in state $x \in \N_+$ is $\mu + \nu$ and the transition probabilities for the jump chain are \[ Q(x, x - 1) = \frac{\nu}{\m...When $k = 1$ , the single-server queue, the exponential parameter in state $x \in \N_+$ is $\mu + \nu$ and the transition probabilities for the jump chain are $Q(x, x - 1) = \frac{\nu}{\mu + \nu}, \; Q(x, x + 1) = \frac{\mu}{\mu + \nu}$ When $k = \infty$ , the infinite server queue, the cases above for $x \ge k$ are vacuous, so the exponential parameter in state $x \in \N$ is $\mu + x \nu$ and the transition probabilities are \[ Q(x, x - 1) = \frac{\nu x}{\mu + \nu…
5: Special Distributions
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions
In this chapter, we study several general families of probability distributions and a number of special parametric families of distributions. Unlike the other expository chapters in this text, the sec...In this chapter, we study several general families of probability distributions and a number of special parametric families of distributions. Unlike the other expository chapters in this text, the sections are not linearly ordered and so this chapter serves primarily as a reference. You may want to study these topics as the need arises.
16.19: Time Reversal in Continuous-Time Chains
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/16%3A_Markov_Processes/16.19%3A_Time_Reversal_in_Continuous-Time_Chains
Then \begin{align*} \P(\hat X_t = y \mid \hat X_s = x, A) & = \frac{\P(\hat X_t = y, \hat X_s = x, A)}{\P(\hat X_s = x, A)} = \frac{\P(X_{h - t} = y, X_{h - s} = x, A)}{\P(X_{h - s} = x, A)} \\ & = \f...Then $\begin{align*} \P(\hat X_t = y \mid \hat X_s = x, A) & = \frac{\P(\hat X_t = y, \hat X_s = x, A)}{\P(\hat X_s = x, A)} = \frac{\P(X_{h - t} = y, X_{h - s} = x, A)}{\P(X_{h - s} = x, A)} \\ & = \frac{\P(A \mid X_{h - t} = y, X_{h - s} = x) \P(X_{h - s} = x \mid X_{h - t} = y) \P(X_{h - t} = y)}{\P(A \mid X_{h - s} = x) \P(X_{h - s} = x)} \end{align*}$ But $A \in \sigma\{X_r: r \in [h - s, h]\}$ and $h - t \lt h - s$ , so by the Markov property for $\bs X$ , \[ \P(A \mid X_{h - t} =…
16.23: Continuous-Time Branching Chains
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/16%3A_Markov_Processes/16.23%3A__Continuous-Time_Branching_Chains
Using the Kolmogorov backward equation we have $\frac{d}{dt} \Phi_t(r) = \sum_{x=0}^\infty r^x \frac{d}{dt} P_t(1, x) = \sum_{x=0}^\infty r^x G P_t(1, x)$ Using the generator above, \[ G P_t(1, x)...Using the Kolmogorov backward equation we have $\frac{d}{dt} \Phi_t(r) = \sum_{x=0}^\infty r^x \frac{d}{dt} P_t(1, x) = \sum_{x=0}^\infty r^x G P_t(1, x)$ Using the generator above, $G P_t(1, x) = \sum_{y = 0}^\infty G(1, y) P_t(y, x) = - \alpha P_t(1, x) + \sum_{k=0}^\infty \alpha f(k) P_t(k, x), \quad x \in \N$ Substituting and using the result above gives \begin{align*} \frac{d}{dt} \Phi_t(r) & = \sum_{x=0}^\infty r^x \left[-\alpha P_t(1, x) + \sum_{k=0}^\infty \alpha f(k) P_t(k, x)\…
5.37: The Wald Distribution
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.37%3A_The_Wald_Distribution
The Wald distribution, named for Abraham Wald, is important in the study of Brownian motion. Specifically, the distribution governs the first time that a Brownian motion with positive drift hits a fix...The Wald distribution, named for Abraham Wald, is important in the study of Brownian motion. Specifically, the distribution governs the first time that a Brownian motion with positive drift hits a fixed, positive value. In Brownian motion, the distribution of the random position at a fixed time has a normal (Gaussian) distribution, and thus the Wald distribution, which governs the random time at a fixed position, is sometimes called the inverse Gaussian distribution.
14: The Poisson Process
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/14%3A_The_Poisson_Process
The Poisson process is one of the most important random processes in probability theory. It is widely used to model random points in time and space, such as the times of radioactive emissions, the arr...The Poisson process is one of the most important random processes in probability theory. It is widely used to model random points in time and space, such as the times of radioactive emissions, the arrival times of customers at a service center, and the positions of flaws in a piece of material. Several important probability distributions arise naturally from the Poisson process—the Poisson distribution, the exponential distribution, and the gamma distribution.
InfoPage
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/00%3A_Front_Matter/02%3A_InfoPage
The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the Californ...The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
5.22: Discrete Uniform Distributions
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.22%3A_Discrete_Uniform_Distributions
The discrete uniform distribution is a special case of the general uniform distribution with respect to a measure, in this case counting measure. The distribution corresponds to picking an element of ...The discrete uniform distribution is a special case of the general uniform distribution with respect to a measure, in this case counting measure. The distribution corresponds to picking an element of S at random. Most classical, combinatorial probability models are based on underlying discrete uniform distributions. The chapter on Finite Sampling Models explores a number of such models.

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