STAT 801

Problems: Assignment 4

Postscript version of these questions

1.

Compute the characteristic function, cumulants and central moments for the Poisson($\lambda$) distribution.

2.

Compute the characteristic function, cumulants and central moments for the Gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$.

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 X+Y/2+Z/4 at 1. You may use the splus function integ.romb (or any other function) found by attaching the directory

/home/math4/lockhart/research/software/quadrature/.Data.

A handout on these functions is available here.

5.

Suppose X is an integer valued random variable. Let $\phi(t)$ be the characteristic function of X.

(a)

Show that

$${\rm P}(X=k)=(2\pi)^{-1}\int_{-\pi}^\pi \phi(t) e^{-itk}\, dt$$

(b)

Suppose further that X is a random variable such that $P(X \mbox{ is even}) = 1$. Show that

$${\rm P}(X=k)=\pi^{-1}\int_{-\pi/2}^{\pi/2} \phi(t) e^{-itk}\, dt$$

6.

Suppose $X_1,\ldots,X_{2n}$ are independent random variables such that ${\rm P}(X_i =1) = {\rm P}(X_i=-1) =0.5$. Prove that as $n \to\infty$

$$(2n)^{1/2}{\rm P}(X_1+\cdots+X_{2n}=0)/\varphi(0) \to 2$$

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

7.

If $(X_1,Y_1),\ldots,(X_n,Y_n)$ are independent bivariate normal random variables find the limiting distribution of $n^{1/2}(r-\rho)$ where r is the sample correlation coefficient and $\rho$ is the population correlation. HINTS: the problem is easier for $\mu_X=\mu_Y=0$ and $\sigma_X=\sigma_Y=1$ so prove that you can assume these values for the parameters without loss of generality. Next remember that the correlation coefficient can be computed from the 5 summary statistics $\bar{X}, \bar{Y}, \bar{X^2}, \bar{Y^2}, \bar{XY}$. Use the central limit theorem (multivariate version) to compute an approximate normal distribution for this vector.