Theory of Statistics - George Mason University

1.36. Consider the the distribution with PDF
p(x) =
c
2x 2 log(|x|) I{±2,±3,...}(x).
Exercises 149
Show that the characteristic function has a finite first derivative at 0, yet
that the first moment does not exist (Zygmund, 1947).
1.37. Write an expression similar to equation (1.94) for the cumulants, if they
exist, in terms of the cumulant-generating function.
1.38. Show that equations (1.97), (1.98), and (1.99) are correct.
1.39. Show that equation (1.105) is correct.
1.40. a) Let X and Y be iid N(0, 1). Work out the PDF of (X − Y ) 2 /Y 2 .
b) Let X1, . . ., Xn and Y1, . . ., Yn
be iid N(0, 1). Work out the PDF of
i (Xi − Yi) 2 / 2
i Yi .
1.41. Show that the distributions of the random variables X and Y in Example
1.7 are the same as, respectively, the ratio of two standard exponential
random variables and the ratio of two standard normal random variables.
1.42. Show that equation (1.130) is correct.
1.43. Stable distributions.
a) Show that an infinitely divisible family of distributions is stable.
b) Show that the converse of the previous statement is not true. (Hint:
Show that the Poisson family is a family of distributions that is infinitely
divisible, but not stable.)
c) Show that the definition of stability based on equation (1.134) is equivalent
to Definition 1.33.
d) Let X, X1, X2 be as in Definition 1.33. Show that Y = X1 − X2 has a
stable distribution, and show that the distribution of Y is symmetric
about 0. (Y has a symmetric stable distribution).
e) Show that the normal family of distributions is stable with characteristic
exponent of 2.
f) Show that the standard Cauchy distribution is stable with characteristic
exponent of 1.
1.44. Prove Theorem 1.25.
1.45. Provide a heuristic justification for equation (1.136).
1.46. Show that the PDF of the joint distribution of all order statistic in equation
(1.138) is equal to the PDF of the joint distribution of all of the
(unordered) random variables, f(xi).
1.47. Show that the Yi in Example 1.19 on page 64 are independent of both
X(1) and X(n).
1.48. a) Let X(1), . . ., X(n) be the order statistics in a sample of size n, let
µ(k:n) = E(X(k:n)), and let X be a random variable with the distribution
of the sample. Show that µ(k:n) exists and is finite if E(X) exists
and is finite.
b) Let n be an odd integer, n = 2k + 1, and consider a sample of size n
from a Cauchy distribution with PDF fX = 1/(π(1+(x −θ) 2 )). Show
that the PDF of X(k+1), the sample median, is
Theory of Statistics c○2000–2013 James E. Gentle