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390Linear Transformation of a Multivariate Normal via MGFLet X \sim N(\boldsymbol , \boldsymbol \Sigma ) be a p-dimensional normal random vector, and let A be a fixed m p matrix. Using the moment-generating function, prove that Z = A X is multivariate normal and determine its mean vector and covariance matrix.概率困难derivation未尝试免费393Sum of Squares of Two Standard NormalsLet X 1, X 2 \sim iid N(0,1). Define R = X 1 2 + X 2 2. (a) Switching to polar coordinates (X 1, X 2) = (r\cos , r\sin ), derive the joint density of (R, \Theta) where R = X 1 2 + X 2 2 and \Theta = \arctan(X 2/X 1). (b) Marginalize over \Theta to find the PDF of R and identify its distribution.概率中等multi part未尝试免费394Ratio of Independent Standard Normals Is CauchyLet X 1, X 2 \sim iid N(0,1). Using the transformation (Y, V) = (X 1/X 2,\, X 2): (a) Derive the joint density f Y,V (y,v). (b) Integrate out V to obtain the marginal PDF of Y = X 1/X 2 and identify the distribution.概率困难derivation未尝试免费395Exponential of a Normal: The Log-Normal DistributionLet X \sim N( , 2) and define Y = e X. (a) Derive the PDF of Y using the change-of-variables formula. (b) Using the MGF of the normal distribution, compute E[Y] and Var (Y). (c) Show that the median of Y is e and explain why E[Y] > median (Y) when > 0.概率困难multi part未尝试免费400Deriving the Fisher F-Distribution from Chi-Squared VariablesLet X \sim \chi 2(m) and Y \sim \chi 2(n) be independent. Define F = X/m Y/n . (a) Using the transformation (F, W) = \bigl( nX mY ,\; Y\bigr), compute the Jacobian and derive the joint density f F,W . (b) Integrate out W to obtain the marginal PDF of F and verify it matches the F(m, n) distribution. (c) Show that E[F] = \dfrac n n-2 for n > 2.概率困难multi part未尝试免费405Joint Distribution of Extremes and the RangeLet X 1, \ldots, X n be iid Uniform (0,1). Let X (1) = \min i X i and X (n) = \max i X i.概率困难multi part未尝试面试订阅410Joint Density and Covariance of Two Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1). Consider the order statistics X (i) and X (j) with 1 \le i < j \le n.概率困难multi part未尝试面试订阅413Beta Distribution of the k-th Uniform Order StatisticLet X 1, \ldots, X n be iid Uniform (0,1). Derive that the k-th order statistic X (k) has the Beta (k, n-k+1) distribution.概率中等derivation未尝试免费414Renyi Representation of Exponential Order-Statistic SpacingsLet X 1, \ldots, X n be iid Exp ( ) and let X (1) \le \cdots \le X (n) be the order statistics. Define the normalized spacings D k = (n-k+1)(X (k) - X (k-1) ) for k = 1, \ldots, n, where X (0) = 0.概率困难multi part未尝试面试订阅418Expected Value of the Second Smallest ExponentialLet X 1, \ldots, X 5 be independent Exp (1) random variables. Derive E[X (2) ].概率中等derivation未尝试免费419Conditional Distribution of the Minimum Given the MaximumLet X 1, \ldots, X n be iid Uniform (0,1) with n \ge 3. Let X (1) and X (n) denote the minimum and maximum.概率困难multi part未尝试面试订阅420Variance of the k-th Uniform Order StatisticLet X 1, \ldots, X n be iid Uniform (0,1). Derive a closed-form expression for Var (X (k) ) for 1 \le k \le n.概率困难derivation未尝试面试订阅425Ratio of the Two Smallest Exponential Order StatisticsLet X 1, X 2 be independent Exp (1) random variables with order statistics X (1) \le X (2) . Define U = X (1) / X (2) .概率困难multi part未尝试面试订阅430Characterization of Memorylessness and the Residual Life ParadoxPart (a): Let X be a continuous, positive random variable satisfying P(X > s + t \mid X > s) = P(X > t) for all s, t 0. Prove that X must be exponentially distributed. Part (b): A lightbulb's lifetime L has CDF F(t) = 1 - 1 2 e -t - 1 2 e -3t for t 0 (a mixture of Exp (1) and Exp (3)). You arrive at a uniformly random time and observe the bulb currently in use. Let R be the residual lifetime of that bulb. Show that E[R] > E[L] and compute both values. Explain why memorylessness breaks down and causes this paradox.概率困难derivation未尝试面试订阅435Uniqueness of Geometric MemorylessnessPart (a): Let N be a positive-integer-valued random variable satisfying P(N > m + n \mid N > m) = P(N > n) for all m, n \in Z 0 . Prove that N must follow a geometric distribution. Part (b): For N \sim Geom (p), compute E[N 2 \mid N > k] using memorylessness and verify that Var (N \mid N > k) = Var (N).概率困难derivation未尝试面试订阅439Sequential Elimination RaceThree players with independent lifetimes X 1 \sim Exp (1), X 2 \sim Exp (2), X 3 \sim Exp (4) compete. The first to "die" is eliminated, then the two survivors continue (by memorylessness, their residual lifetimes are fresh exponentials with the same rates). The second to die is eliminated, and the last survivor wins. (a) Find the probability that the elimination order is X 3, X 1, X 2 (i.e., player 3 dies first, then player 1, then player 2). (b) Find the expected total time until only one player remains.概率困难multi part未尝试面试订阅444Full Ordering Probability for Four Competing ExponentialsFour independent exponential random variables X 1 \sim Exp (1), X 2 \sim Exp (2), X 3 \sim Exp (3), X 4 \sim Exp (6) represent task completion times. Using iterated applications of the memoryless property, find P(X 4 < X 3 < X 2 < X 1) — the probability that the tasks complete in the specific order 4, 3, 2, 1.概率困难数值题未尝试面试订阅445Memorylessness Breaks for Exponential MixturesLet X have the mixture density f(x) = 1 2 e -x + 5 2 e -5x for x 0 (a 50 - 50 mixture of Exp (1) and Exp (5)). (a) Compute P(X > s + t \mid X > s) as a function of s and t, and show it depends on s (i.e., the memoryless property fails). (b) Evaluate P(X > 2 \mid X > 1) and compare with P(X > 1). (c) Interpret: given that X has survived past s, how does the conditional distribution change as s increases?概率困难multi part未尝试面试订阅450Head Start in an Exponential RaceLet X \sim Exp ( ) and Y \sim Exp ( ) be independent. Player A finishes at time X and player B finishes at time Y + c where c > 0 is a head start for player A (player B starts c time units later). (a) Derive P(X < Y + c) — the probability that A finishes before B. (b) Show that as c 0, the result recovers the standard competing-exponentials formula. (c) Evaluate for = 3, = 2, c = 1 and interpret how the head start affects A's winning probability.概率困难multi part未尝试面试订阅454Berry-Esseen Bound for a Skewed Bernoulli SumLet X 1, X 2, \ldots, X n be i.i.d.\ Bernoulli (p) with p = 0.01 and n = 10 , 000. Define S n = \sum i=1 n X i. **(a)** Using the CLT, approximate P(S n \le 80). **(b)** The Berry-Esseen theorem states that \sup x |P(Z n \le x) - \Phi(x)| \le C\, 3 n , where Z n = (S n - n )/( n ), = E[|X 1 - | 3], and C \le 0.4748. Compute the Berry-Esseen bound on the approximation error in part (a). You may use \Phi(-2) \approx 0.0228.概率中等derivation未尝试免费