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068Independence Is Not Hereditary to Sub-EventsLet \Omega = \ 1,2,3,4\ with uniform probability. Define A = \ 1,2\ and B = \ 1,3\ . (a) Verify that A and B are independent. (b) Let A 1 = \ 1\ \subseteq A. Determine whether A 1 and B are independent. (c) State a general principle: if A \perp\!\!\perp B and A 1 \subseteq A, does A 1 \perp\!\!\perp B necessarily hold?概率中等derivation未尝试免费098The Will Rogers Paradox (Stage Migration)Group A consists of the values \ 1, 2, 3, 9\ and Group B consists of \ 5, 6, 7, 8\ . (a) Compute the mean of each group. (b) Now move the element 5 from Group B to Group A. Recompute the mean of each group. Verify that both means have increased. (c) The overall mean across all 8 values is unchanged. Prove the following general statement: if an element is moved from group B to group A, and the element's value x satisfies A < x < B (where A , B are the pre-move group means), then both post-move group means strictly exceed their pre-move values. (d) A medical study reports that a new diagnostic technology has improved average survival time in both Stage I and Stage II cancer patients, yet overall survival is unchanged. Explain how this is possible without any actual improvement in treatment.概率中等数值题未尝试免费219Distribution of the Maximum of Independent Geometric Random VariablesLet X 1, X 2, \ldots, X n be independent Geometric (p) random variables with P(X i = k) = (1-p) k-1 p for k = 1, 2, \ldots Define M = \max(X 1, \ldots, X n). (a) Show that P(M \le m) = [1 - (1-p) m] n for m = 1, 2, \ldots (b) Derive P(M = m) from the CDF. (c) Express E[M] as an infinite series using the tail-sum formula E[M] = \sum m=0 P(M > m). Simplify to: E[M] = \sum m=0 [1 - (1 - (1-p) m) n ]. (d) For the special case n = 2, p = 1/2, compute P(M = 1), P(M = 2), P(M = 3) and verify they sum to nearly 1. Compute E[M] exactly by evaluating the series. (e) For general n and small p, argue heuristically that E[M] \approx \ln n p by comparing to the continuous exponential analogue.概率困难derivation未尝试免费250The Folded Normal Distribution: PDF and Moments of |X|Let X \sim N( , 2) with 0. Define Y = |X|. (a) Derive the PDF of Y for y 0 by writing P(Y y) = P(-y X y) and differentiating. (b) Show that when = 0 the PDF simplifies to the half-normal distribution f Y(y) = 2 \, e -y 2/(2 2) for y 0. (c) Derive E[Y] and Var (Y) for general , . Express your answer using the standard normal PDF \phi and CDF \Phi. (d) Compute E[Y] and Var (Y) numerically for = 1, = 1.概率困难derivation未尝试免费1426Cubic Residue of an Exponential RemainderCompute lim x->0 [e (3x) - 1 - 3x - (9/2)x 2] / x 3.数学简单数值题未尝试免费1427Third-Order Logarithmic CancellationCompute lim x->0 [ln(1+2x) - 2x + 2x 2] / x 3.数学简单数值题未尝试免费1429Square-Root Cubic CoefficientCompute lim x->0 [sqrt(1+4x) - 1 - 2x + 2x 2] / x 3.数学中等数值题未尝试免费1431Half-Power Expansion With Cubic TermCompute lim x->0 [(1+5x) (1/2) - 1 - (5/2)x + (25/8)x 2] / x 3.数学简单数值题未尝试免费1432Rational Blow-Up CancellationCompute lim x->0 [(1-2x) (-1) - 1 - 2x] / x 2.数学中等数值题未尝试免费1434Tangent Minus IdentityCompute lim x->0 (tan x - x) / x 3.数学中等数值题未尝试免费1436Cosine Defect at Double FrequencyCompute lim x->0 [1 - cos(2x)] / x 2.数学简单数值题未尝试免费1437Cube-Root LinearizationCompute lim x->0 [ (1+2x) (1/3) - 1 ] / x.数学简单数值题未尝试免费1441Trigonometric Asymptotic 1Compute lim (x->0) [sin(2x + 7x 2) - 2x] / x 2.数学简单数值题未尝试免费1442Exponential Minus CosineCompute lim x->0 [e x - cos x - x] / x 2.数学中等数值题未尝试免费1446Mixed-Expansion Limit 1Compute lim (x->0) [((e x - 1) - ln(1+x)) / x 2].数学简单数值题未尝试免费1447Mixed-Expansion Limit 2Compute lim (x->0) [((sqrt(1+x) - sqrt(1-x)) - x) / x 3].数学简单数值题未尝试免费1449Mixed-Expansion Limit 4Compute lim (x->0) [e x cos(x) - 1 - x] / x 2.数学中等数值题未尝试免费1542Sherman-Morrison Vector Update B 2Suppose A is invertible, A (-1)b = [2, 4] T, A (-1)u = [1, -1] T, v T A (-1)b = -3, and v T A (-1)u = 1/4. What is (A + u v T) (-1) b?数学中等数值题未尝试免费1546Equal-Factor Determinant 1What is det(3 I + 2 11 T) as an n=4 dimensional matrix?数学简单数值题未尝试免费1549Ridge-Plus-Ones Determinant 4What is det(4 I + 1/2 11 T) as an n=5 dimensional matrix?数学中等数值题未尝试免费