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224Compound Poisson Distribution: MGF and MomentsLet N \sim Poisson ( ) and let X 1, X 2, \ldots be iid discrete random variables (independent of N) with PMF P(X i = j) = p j for j = 1, 2, \ldots and MGF M X(t) = E[e tX 1 ]. Define the **compound Poisson** sum S = \sum i=1 N X i (with S = 0 when N = 0). (a) Derive the MGF of S. Show that M S(t) = \exp\!\big( (M X(t) - 1)\big). (b) Use the MGF to derive E[S] and Var (S). Express your answers in terms of , E[X 1], and Var (X 1). (c) Alternatively, derive E[S] and Var (S) using the tower property (law of total expectation) and the Eve's law (law of total variance), conditioning on N. (d) **Application:** An insurance company receives claims at a Poisson rate of = 10 per day. Each claim size is \1000 with probability 0.6 or \5000 with probability 0.4, independently. Find E[S] and Var (S) for the total daily claims S, and compute the standard deviation.概率困难derivation未尝试免费225Minimum of Independent Geometric Random VariablesLet X 1, X 2, \ldots, X n be independent, each X i \sim Geometric (p i) with P(X i = k) = (1-p i) k-1 p i for k = 1, 2, \ldots (the "number of trials until first success" convention). Define M = \min(X 1, \ldots, X n). (a) Show that P(M > k) = \prod i=1 n (1-p i) k for k = 0, 1, 2, \ldots (b) Prove that M \sim Geometric \! (1 - \prod i=1 n (1-p i) ). State the PMF of M explicitly. (c) For the iid case p i = p for all i: express E[M] and Var (M) in terms of n and p, and verify that E[M] 1 as n . (d) **Application:** Five independent traders each attempt to fill an order on any given day with probability 0.3. What is the expected number of days until the first fill occurs? What is the probability that no fill occurs in the first 3 days? (e) Show that P(X j = M \mid M = m) depends on j (when p i are not all equal) and compute this probability for n = 2, p 1 = 0.3, p 2 = 0.5, m = 2.概率困难derivation未尝试免费226Mean and Variance of the Continuous Uniform DistributionLet X \sim Uniform (a, b). Starting from the PDF f(x) = 1 b-a for x \in [a, b], derive E[X] and Var (X).概率简单derivation未尝试免费227Memorylessness of the Exponential DistributionLet X \sim Exponential ( ). Prove that P(X > s + t \mid X > s) = P(X > t) for all s, t 0. Then show the exponential is the only continuous distribution with this property.概率简单derivation未尝试免费228Moment Generating Function of the Gamma DistributionLet X \sim Gamma ( , ) with PDF f(x) = \Gamma( ) x -1 e - x for x > 0. Derive the moment generating function M X(t) = E[e tX ] for t < . Then use M X(t) to compute E[X] and Var (X).概率中等derivation未尝试免费229Standard Normal Tail and Interval ProbabilitiesLet Z \sim N(0,1). Using \Phi(1) \approx 0.8413 and \Phi(2) \approx 0.9772, compute: (a) P(|Z| > 2) and (b) P(1 < Z < 2). Explain each step using symmetry of the normal distribution.概率中等数值题未尝试免费230Beta Distribution: Derivation, Moments, and ConnectionsConsider the Beta distribution with PDF f(x) = \Gamma( + ) \Gamma( )\Gamma( ) x -1 (1-x) -1 for x \in (0,1). (a) Verify that f integrates to 1 by showing \int 0 1 x -1 (1-x) -1 \,dx = \Gamma( )\Gamma( ) \Gamma( + ) using the integral representation of the Gamma function. (b) Derive E[X] = + and Var (X) = ( + ) 2( + +1) . (c) Show that Beta (1,1) reduces to Uniform (0,1). (d) If Y 1 \sim Gamma ( , 1) and Y 2 \sim Gamma ( , 1) are independent, explain (without full proof) why Y 1 Y 1+Y 2 \sim Beta ( , ).概率困难derivation未尝试免费231CDF and Tail Probability of the Exponential DistributionLet X \sim Exponential ( ) with PDF f(x) = e - x for x 0. (a) Derive the CDF F(x) = P(X x). (b) If = 1 2 , compute P(X > 3).概率简单数值题未尝试免费232Sum of Two Independent Exponentials via ConvolutionLet X 1, X 2 be independent, each distributed Exponential ( ). Using the convolution formula, derive the PDF of Y = X 1 + X 2 and identify the resulting distribution.概率中等derivation未尝试免费233Mean and Variance of the Lognormal DistributionLet X be a lognormal random variable, meaning \ln X \sim N( , 2). Using the moment generating function of the normal distribution, derive E[X] and Var (X).概率中等derivation未尝试免费235Chi-Squared Distribution from First PrinciplesLet Z 1, \ldots, Z n be i.i.d. N(0,1) random variables and define Q = \sum i=1 n Z i 2. (a) Derive the PDF of Z 1 2 using the change-of-variables technique for transformations of continuous random variables. (b) Show that Z 1 2 \sim Gamma ( 1 2 , 1 2 ) by matching the PDF from (a) to the Gamma family. (c) Using the fact that the sum of independent Gamma random variables with the same rate parameter is Gamma, state the distribution of Q. (d) Derive E[Q] and Var (Q).概率困难derivation未尝试免费236Median of the Weibull DistributionThe Weibull distribution with shape k > 0 and scale > 0 has CDF F(x) = 1 - e -(x/ ) k , \quad x 0. (a) Derive a closed-form expression for the median of this distribution. (b) Compute the median when k = 2 and = 3.概率简单数值题未尝试免费237Mean and Variance of the Gamma Distribution via IntegrationLet X \sim Gamma ( , ) with PDF f(x) = \Gamma( ) x - 1 e - x for x > 0. (a) Derive E[X] by direct integration, using the fact that \int 0 \Gamma( ) x - 1 e - x \,dx = 1 for any > 0. (b) Derive E[X 2] similarly and compute Var (X).概率简单derivation未尝试免费238Non-Existence of the Mean for the Cauchy DistributionThe standard Cauchy distribution has PDF f(x) = 1 (1 + x 2) for x \in (- , ). (a) Show that E[|X|] does not exist by proving the integral \int 0 x (1+x 2) \,dx diverges. (b) What does this imply about the applicability of the Law of Large Numbers to the sample mean X n = 1 n \sum i=1 n X i when X i are i.i.d. Cauchy?概率中等derivation未尝试免费239Ratio of Independent Standard Normals is CauchyLet Z 1, Z 2 be independent N(0,1) random variables. Define R = Z 1 / Z 2. (a) Write the joint PDF of (Z 1, Z 2) and use the transformation (Z 1, Z 2) \mapsto (R, Z 2) = (Z 1/Z 2,\, Z 2) to derive the joint PDF of (R, Z 2). (b) Integrate out Z 2 to obtain the marginal PDF of R. (c) Identify the distribution of R.概率困难derivation未尝试免费240Beta Distribution and the Beta Function from Independent GammasLet X \sim Gamma ( , 1) and Y \sim Gamma ( , 1) be independent. (a) Define U = X X+Y and V = X + Y. Compute the Jacobian of the transformation (X, Y) \mapsto (U, V). (b) Derive the joint PDF of (U, V) and show that U and V are independent. (c) Identify the marginal distribution of U and prove that B( , ) = \Gamma( )\,\Gamma( ) \Gamma( + ) , where B( , ) = \int 0 1 u -1 (1-u) -1 \,du.概率困难derivation未尝试免费241Tail Probability and Conditional Expectation of the Pareto DistributionThe Pareto distribution with shape > 0 and scale x m > 0 has PDF f(x) = \, x m x +1 , \quad x x m. (a) Derive P(X > t) for t x m. (b) Derive E[X \mid X > t] for > 1 and t x m. (c) Compute both quantities for = 3, x m = 1, t = 2.概率简单数值题未尝试免费242Distribution of the Sum of Two Independent UniformsLet X and Y be independent Uniform (0, 1) random variables. Define S = X + Y. (a) Using the convolution formula f S(s) = \int - f X(t)\, f Y(s - t)\, dt, derive the PDF of S for all s \in R . (b) Sketch the PDF and identify the distribution by name. (c) Compute P(S > 1.5).概率中等derivation未尝试免费243Inverse Transform Sampling for the Exponential DistributionLet U \sim Uniform (0, 1). (a) For a continuous random variable X with strictly increasing CDF F, prove that F(X) \sim Uniform (0,1). (b) Use part (a) to argue that X = F -1 (U) has CDF F. (c) For X \sim Exp ( ) with CDF F(x) = 1 - e - x (x 0), derive F -1 explicitly and write a one-line formula for generating exponential samples from uniform samples.概率中等derivation未尝试免费244Distribution of the Square of a Standard Normal via Change of VariablesLet X \sim N(0, 1). (a) The transformation Y = X 2 is not monotone. Using the CDF method, derive f Y(y) by first computing F Y(y) = P(X 2 y) and then differentiating. (b) Alternatively, apply the non-monotone change-of-variables formula: if Y = g(X) and g has exactly two branches x 1(y) and x 2(y) satisfying g(x i) = y, then f Y(y) = \sum i=1 2 f X(x i(y))\, | dx i dy |. Verify you get the same answer. (c) Identify the resulting distribution and express it as a Gamma distribution.概率困难derivation未尝试免费