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090Borel's Paradox: Conditioning on Measure-Zero EventsLet (\Theta, \Phi) be uniformly distributed on the unit sphere S 2, where \Theta \in [0, 2 ) is the longitude and \Phi \in [0, ] is the colatitude, with joint density f( , \phi) = 1 4 \sin \phi. (a) Compute the conditional distribution of \Phi given \Theta = 0 by treating \Theta as the conditioning variable (i.e., compute f(\phi \mid = 0)). (b) Now reparameterize: let X = \cos(\Theta) \sin(\Phi), Y = \sin(\Theta) \sin(\Phi), Z = \cos(\Phi). The great circle \ \Theta = 0\ can equivalently be described as \ Y = 0, X 0\ . Compute the conditional distribution of \Phi given Y = 0 and X > 0. Is it the same as in part (a)? (c) Explain why the two answers differ. What does this tell us about the meaning of 'conditioning on a measure-zero event,' and what is the correct mathematical framework for resolving this ambiguity?概率困难derivation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246Rayleigh Distribution from Independent NormalsLet X and Y be independent N(0, 2) random variables. Define R = X 2 + Y 2 . (a) Using the CDF method, derive the PDF of R for r 0. (b) Identify the distribution by name and compute E[R]. (c) Evaluate E[R] and Var (R) numerically for = 1.概率简单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377Distribution of the Exponential of a Uniform via JacobianLet X \sim Uniform (0,1). Use the change-of-variables (Jacobian) formula to find the PDF of Y = e X.概率简单derivation未尝试免费379Distribution and Mean of the Maximum of Two ExponentialsLet X and Y be independent Exp (1) random variables. Define M = \max(X, Y). (a) Derive the PDF of M. (b) Compute E[M].概率中等数值题未尝试免费380Distribution of the Ratio of Two Independent ExponentialsLet X and Y be independent Exp (1) random variables. Define R = X/Y. (a) Using the transformation (R, S) = (X/Y,\, Y), compute the joint density f R,S via the Jacobian and then marginalize over S to find the PDF of R. (b) Identify f R as a named distribution and verify by computing P(R \le 1) using a symmetry argument.概率困难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382Square of a Standard Normal Is Chi-Squared(1)Let X \sim N(0,1). Using the CDF method, derive the PDF of Y = X 2 and identify the resulting distribution.概率中等derivation未尝试免费383Reciprocal Transform of an Exponential VariableLet X \sim Exp (1). Define Y = \dfrac 1 1 + X . (a) Use the change-of-variables formula to derive the PDF of Y. (b) Compute E[Y].概率中等derivation未尝试免费384Distribution of the Product of Two Independent UniformsLet X and Y be independent Uniform (0,1) random variables. Using the transformation (W, V) = (XY,\, Y), derive the PDF of W = XY.概率中等derivation未尝试免费385Box-Muller Transform: From Uniforms to Independent NormalsLet U 1, U 2 be independent Uniform (0,1) random variables. Define Z 1 = -2\ln U 1 \,\cos(2 U 2), \qquad Z 2 = -2\ln U 1 \,\sin(2 U 2). (a) Compute the Jacobian of the inverse transformation from (Z 1, Z 2) back to (U 1, U 2). (b) Show that Z 1 and Z 2 are independent N(0,1) random variables.概率困难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