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337Robust Variance of a Difference of Independent VariablesLet X and Y be independent random variables with Var (X) = 4 and Var (Y) = 9. A student claims that SD (X - Y) = SD (X) - SD (Y) = 2 - 3 = -1. Find the correct value of Var (X - Y) and SD (X - Y), and explain the student's error. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费376Distribution of the Cube of a Uniform Random VariableLet X \sim Uniform (0,1). Using the CDF method, derive the PDF of Y = X 3.概率简单derivation未尝试免费381Negative Log of a Uniform Yields an ExponentialLet X \sim Uniform (0,1). Using the CDF method, derive the PDF of Y = -\ln X and identify the resulting distribution.概率简单derivation未尝试免费386Affine Transformation of a Normal Random VariableLet X \sim N( , 2) with a 0 and b \in R . Using the Jacobian formula, show that Y = aX + b is normally distributed and state its parameters.概率简单derivation未尝试免费391Square Root of an Exponential Random VariableLet X \sim Exp (1). Use the change-of-variables formula to derive the PDF of Y = X and identify the resulting distribution.概率简单derivation未尝试免费396Distribution of the Maximum of n Uniform Random VariablesLet X 1, \ldots, X n \sim iid Uniform (0,1). Derive the CDF and PDF of M = \max(X 1, \ldots, X n).概率简单derivation未尝试免费397Reciprocal of a Uniform Random VariableLet X \sim Uniform (0,1). Use the change-of-variables formula to derive the PDF of Y = 1/X. Determine whether E[Y] is finite.概率简单derivation未尝试免费399Absolute Value of a Standard Normal: The Half-Normal DistributionLet X \sim N(0,1) and define Y = |X|. (a) Derive the PDF of Y using the CDF method (note that Y = |X| is not monotone). (b) Compute E[Y] and Var (Y).概率中等multi part未尝试免费406Second Order Statistic from Five UniformsLet X 1, \ldots, X 5 be independent Uniform (0,1) random variables and let X (2) denote the second smallest. Find E[X (2) ].概率简单数值题未尝试免费411Probability Involving the Third Order StatisticLet X 1, X 2, X 3, X 4 be independent Uniform (0,1) random variables. Compute P(X (3) < 0.5), where X (3) is the third smallest.概率简单数值题未尝试免费412Expected Minimum of Five UniformsLet X 1, \ldots, X 5 be independent Uniform (0,1) random variables. Find E[X (1) ], the expected value of the minimum.概率简单数值题未尝试免费416Second Smallest of Four UniformsLet X 1, X 2, X 3, X 4 be independent Uniform (0,1) random variables. Compute E[X (2) ].概率简单数值题未尝试免费421CDF of the Minimum of Four UniformsLet X 1, X 2, X 3, X 4 be independent Uniform (0,1) random variables. Derive the CDF and PDF of X (1) = \min(X 1, X 2, X 3, X 4).概率简单derivation未尝试免费422Expected Maximum of Two UniformsLet X 1, X 2 be independent Uniform (0,1) random variables. Compute E[\max(X 1, X 2)].概率简单数值题未尝试免费427Conditional Expectation via MemorylessnessLet X \sim Exp (2). Using the memoryless property, compute E[X \mid X > 3].概率简单数值题未尝试免费431Geometric Survival Past a ThresholdLet X \sim Geom (1/4) (number of trials until first success). Using the memoryless property of the geometric distribution, compute (i) E[X \mid X > 5] and (ii) P(X > 8 \mid X > 5).概率简单数值题未尝试免费436Direct Application of Exponential MemorylessnessA radioactive atom has a lifetime X \sim Exp (1/2). Given that the atom has survived past time t = 3, what is the probability it survives past time t = 7?概率简单数值题未尝试免费437Fresh Start After a Losing StreakYou flip a coin with P( heads ) = 1/3 until you get heads. Given that the first 8 flips were all tails, what is the expected total number of additional flips needed (starting from flip 9)?概率简单数值题未尝试免费441Minimum of Three Identical ExponentialsLet X 1, X 2, X 3 be independent, each Exp (4). Find the distribution of M = \min(X 1, X 2, X 3) and compute E[M]. Then verify: given that M > 2, use memorylessness to find E[M \mid M > 2].概率简单数值题未尝试免费446Geometric Coupon Collector's Conditional SurvivalYou roll a fair die repeatedly until you roll a 6. Let N be the number of rolls needed. Given that you have not yet rolled a 6 in the first 5 rolls, what is the probability that you will need more than 10 rolls total?概率简单数值题未尝试免费