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- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/01%3A_Probability_Systems/1.03%3A_Interpretations(IF5): \(I_{A \cup B} = I_A + I_B - I_A I_B = \text{min }{I_A, I_B}\) (the maximum rule extends to any class) The maximum rule follows from the fact that \(\omega\) is in the union iff it is in any on...(IF5): \(I_{A \cup B} = I_A + I_B - I_A I_B = \text{min }{I_A, I_B}\) (the maximum rule extends to any class) The maximum rule follows from the fact that \(\omega\) is in the union iff it is in any one or more of the events in the union iff any one or more of the individual indicator function has value one iff the maximum is one.
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/12%3A_Variance_Covariance_and_Linear_Regression/12.01%3A_Variancejdemo1 % Call for data jcalc % Set up Enter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X X Enter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t...jdemo1 % Call for data jcalc % Set up Enter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X X Enter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t, u, and P G = t.^2 + 2*t.*u - 3*u; % calcculation of matrix of [g(t_i, u_j)] EG = total(G.*P) % Direct calculation of E[g(X,Y)] EG = 3.2529 VG = total(G.^.*P) - EG^2 % Direct calculation of Var[g(X,Y)] VG = 80.2133 [Z,PZ] = csort(G,P); % Determination of distribution for Z EZ = Z*PZ' % E[Z] from d…
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/17%3A_Appendices/17.05%3A_Appendix_E_to_Applied_Probability_-_Properties_of_Mathematical_Expectation(E19): Special case of the Radon-Nikodym theorem If \(g(Y)\) is integrable and \(X\) is a random vector, then there exists a real-valued Borel function \(e(\cdot)\), defined on the range of \(X\), uni...(E19): Special case of the Radon-Nikodym theorem If \(g(Y)\) is integrable and \(X\) is a random vector, then there exists a real-valued Borel function \(e(\cdot)\), defined on the range of \(X\), unique a.s. \([P_X]\), such that \(E[I_M(X) g(X)] = E[I_M (X) e(X)]\) for all Borel sets \(M\) on the codomain of \(X\). \(\sum_{n = 0}^{\infty} P(X \ge n + 1) \le E[X] \le \sum_{n = 0}^{\infty} P(X \ge n) \le N \sum_{k = 0}^{\infty} P(X \ge kN)\), for all \(N \ge 1\)
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/00%3A_Front_Matter/03%3A_PrefaceCONNEXIONS The development and organization of the CONNEXIONS modules has been achieved principally by two people: C.S.(Sid) Burrus a former student and later a faculty colleague, then Dean of Enginee...CONNEXIONS The development and organization of the CONNEXIONS modules has been achieved principally by two people: C.S.(Sid) Burrus a former student and later a faculty colleague, then Dean of Engineering, and most importantly a long time friend; and Daniel Williamson, a music major whose keyboard skills have enabled him to set up the text (especially the mathematical expressions) with great accuracy, and whose dedication to the task has led to improvements in presentation.
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/00%3A_Front_Matter/02%3A_InfoPageThe 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.
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/15%3A_Random_Selection/15.03%3A_Problems_on_Random_SelectiongN = (1/20)*[0 ones(1,20)]; gY = [5/6 1/6]; gend Do not forget zero coefficients for missing powers Enter gen fn COEFFICIENTS for gN gN Enter gen fn COEFFICIENTS for gY gY Results are in N, PN, Y, PY,...gN = (1/20)*[0 ones(1,20)]; gY = [5/6 1/6]; gend Do not forget zero coefficients for missing powers Enter gen fn COEFFICIENTS for gN gN Enter gen fn COEFFICIENTS for gY gY Results are in N, PN, Y, PY, D, PD, P May use jcalc or jcalcf on N, D, P To view the distribution, call for gD.
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/01%3A_Probability_Systems/1.04%3A_Problems_on_Probability_Systems\(A = E_1 \bigvee E_1^c E_2^c E_3 \bigvee E_1^c E_2^c E_3^c E_4^c E_5 \bigvee E_1^c E_2^c E_3^c E_4^c E_5^c E_6^c E_7 \bigvee E_1^c E_2^c E_3^c E_4^c E_5^c E_6^c E_7^c E_8^c E_9\) \(B = E_1^c E_2 \big...\(A = E_1 \bigvee E_1^c E_2^c E_3 \bigvee E_1^c E_2^c E_3^c E_4^c E_5 \bigvee E_1^c E_2^c E_3^c E_4^c E_5^c E_6^c E_7 \bigvee E_1^c E_2^c E_3^c E_4^c E_5^c E_6^c E_7^c E_8^c E_9\) \(B = E_1^c E_2 \bigvee E_1^c E_2^c E_3^c E_4 \bigvee E_1^c E_2^c E_3^c E_4^c E_5^c E_6 \bigvee E_1^c E_2^c E_3^c E_4^c E_5^c E_6^c E_7^c E_8 \bigvee E_1^c E_2^c E_3^c E_4^c E_5^c E_6^c E_7^c E_8^c E_9^c E_{10}\) \(P(A^c B) = P(B) - P(AB) = 0\) \(P(A^c B^c) = P(A^c) - P(A^c B) = 0.5\) \(P(A \cup B) = 0.5\)
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/02%3A_Minterm_Analysis/2.01%3A_MintermsIf we partition an event F into component events whose probabilities can be determined, then the additivity property implies the probability of F is the sum of these component probabilities. Frequ...If we partition an event F into component events whose probabilities can be determined, then the additivity property implies the probability of F is the sum of these component probabilities. Frequently, the event F is a Boolean combination of members of a finite class. For each such finite class, there is a fundamental partition determined by the class. The members of this partition are called minterms.
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/13%3A_Transform_Methods/13.02%3A_Convergence_and_the_Central_Limit_TheoremIn the statistics of large samples, the sample average is a constant times the sum of the random variables in the sampling process . Thus, for large samples, the sample average is approximately normal...In the statistics of large samples, the sample average is a constant times the sum of the random variables in the sampling process . Thus, for large samples, the sample average is approximately normal—whether or not the population distribution is normal. In the case of sample average, the “closeness” to a limit is expressed in terms of the probability that the observed value \(X_n (\omega)\) should lie close the the value \(X(\omega)\) of the limiting random variable.
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/12%3A_Variance_Covariance_and_Linear_Regression/12.02%3A_Covariance_and_the_Correlation_CoefficientReference to Figure 12.2.1 shows this is the average of the square of the distances of the points \((r, s) = (X^*, Y^*) (\omega)\) from the line \(s = r\) (i.e. \(1 - \rho\) is proportional to the var...Reference to Figure 12.2.1 shows this is the average of the square of the distances of the points \((r, s) = (X^*, Y^*) (\omega)\) from the line \(s = r\) (i.e. \(1 - \rho\) is proportional to the variance abut the \(\rho = 1\) line and \(1 + \rho\) is proportional to the variance about the \(\rho = -1\) line. \(\rho = 0\) iff the variances about both are the same.
- https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_(Pfeiffer)/05%3A_Conditional_Independence/5.01%3A_Conditional_IndependenceIf any of the pairs {\(A, B\)}, {\(A, B^c\)}, {\(A^c, B\)} or {\(A^c, B^c\)} is conditionally independent, given C, then so are the others. Suppose D is the event the patient has the disease, A is the...If any of the pairs {\(A, B\)}, {\(A, B^c\)}, {\(A^c, B\)} or {\(A^c, B^c\)} is conditionally independent, given C, then so are the others. Suppose D is the event the patient has the disease, A is the event the first test is positive (indicates the conditions associated with the disease) and B is the event the second test is positive.
