3. X is the random variable for claim sizes. Given A, X follow a single-parameter Pareto...
Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b> 0. (a) Find the cumulative distribution function of Y = XI(X < b} (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
I. Let X be a random sample from an exponential distribution with unknown rate parameter θ and p.d.f (a) Find the probability of X> 2. (b) Find the moment generating function of X, its mean and variance. (c) Show that if X1 and X2 are two independent random variables with exponential distribution with rate parameter θ, then Y = X1 + 2 is a random variable with a gamma distribution and determine its parameters (you can use the moment generating...
Q1. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function F and density f. Let b>0. (a) Write the forinula for E(X b)+1. (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx. Let b > 0 a) Find the cumulative distribution function ofY -XKX< (b) Apply the general formula fron (a) to exponential distribution with parameter > 0.
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
Two samples Xi and X2 are taken from an exponential random variable X with unknown parameter θ; that is. r (a r >0. We propose two estimators for θ in the forms 4 In terms of unbiasedness and minimum variance, which one is the better of the two?
Q3. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b>0 (a) Find the cumulative distribution function of Y- (X -b)+ (b) Apply the general formula from (a) to Pareto distribution with parameter a > 0. Hint: Consider separately cases b e (0, 1] and b> 1.
1. [8 points] Suppose Xi... Xn is a random sample from a Pareto distribution with the density If x > 1 otherwise, where ? > 1, Find the method of moments estimator of ?.