STAT 801

Problems: Assignment 4

1. Compute the characteristic function, cumulants and central moments for the Poisson() distribution.

2. Compute the characteristic function, cumulants and central moments for the Gamma distribution with shape parameter and scale parameter .

3. Develop explicit formulas for the saddlepoint approximation to the density of the mean of a sample of size n from the exponential distribution. Compare the results with the true Gamma density.

4. Suppose X, Y and Z are independent standard exponentials. Use numerical Fourier inversion of the characteristic function to compute the density of at 1. You may use the splus function integ.romb (or any other function) found by attaching the directory

   /home/math/lockhart/research/software/quadrature/.Data.

5. Suppose X is an integer valued random variable. Let be the characteristic function of X. Show that

   

6. Suppose are independent random variables such that . Prove that

   

   where is the standard normal density. You should use the previous problem and Taylor expansion of the characteristic function around 0. Also do the same thing using Sterling's formula.

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