Find the generating functions, both ordinary [math]h(z)[/math] and moment [math]g(t)[/math], for the following discrete probability distributions.
- The distribution describing a fair coin.
- The distribution describing a fair die.
- The distribution describing a die that always comes up 3.
- The uniform distribution on the set [math]\{n,n+1,n+2,\ldots,n+k\}[/math].
- The binomial distribution on [math]\{n,n+1,n+2,\ldots,n+k\}[/math].
- The geometric distribution on [math]\{0,1,2,\ldots,\}[/math] with [math]p(j) = 2/3^{j + 1}[/math].
Let [math]p[/math] be a probability distribution on [math]\{0,1,2\}[/math] with moments [math]\mu_1 = 1[/math], [math]\mu_2 = 3/2[/math].
- Find its ordinary generating function [math]h(z)[/math].
- Using (a), find its moment generating function.
- Using (b), find its first six moments.
- Using (a), find [math]p_0[/math], [math]p_1[/math], and [math]p_2[/math].
In Exercise, the probability distribution is completely determined by its first two moments. Show that this is always true for any probability distribution on [math]\{0,1,2\}[/math]. Hint: Given [math]\mu_1[/math] and [math]\mu_2[/math], find [math]h(z)[/math] as in Exercise and use [math]h(z)[/math] to determine [math]p[/math].
Let [math]p[/math] and [math]p'[/math] be the two distributions
- Show that [math]p[/math] and [math]p'[/math] have the same first and second moments, but not the same third and fourth moments.
- Find the ordinary and moment generating functions for [math]p[/math] and [math]p'[/math].
Let [math]p[/math] be the probability distribution
and let [math]p_n = p * p * \cdots * p[/math] be the [math]n[/math]-fold convolution of [math]p[/math] with itself.
- Find [math]p_2[/math] by direct calculation (see Definition).
- Find the ordinary generating functions [math]h(z)[/math] and [math]h_2(z)[/math] for [math]p[/math] and [math]p_2[/math], and verify that [math]h_2(z) = (h(z))^2[/math].
- Find [math]h_n(z)[/math] from [math]h(z)[/math].
- Find the first two moments, and hence the mean and variance, of [math]p_n[/math] from [math]h_n(z)[/math]. Verify that the mean of [math]p_n[/math] is [math]n[/math] times the mean of [math]p[/math].
- Find those integers [math]j[/math] for which [math]p_n(j) \gt 0[/math] from [math]h_n(z)[/math].
Let [math]X[/math] be a discrete random variable with values in [math]\{0,1,2,\ldots,n\}[/math] and moment generating function [math]g(t)[/math]. Find, in terms of [math]g(t)[/math], the generating functions for
- [math]-X[/math].
- [math]X + 1[/math].
- [math]3X[/math].
- [math]aX + b[/math].
Let [math]X_1[/math], [math]X_2[/math], ..., [math]X_n[/math] be an independent trials process, with values in [math]\{0,1\}[/math] and mean [math]\mu = 1/3[/math]. Find the ordinary and moment generating functions for the distribution of
- [math]S_1 = X_1[/math]. Hint: First find [math]X_1[/math] explicitly.
- [math]S_2 = X_1 + X_2[/math].
- [math]S_n = X_1 + X_2 +\cdots+ X_n[/math].
Let [math]X[/math] and [math]Y[/math] be random variables with values in [math]\{1,2,3,4,5,6\}[/math] with distribution functions [math]p_X[/math] and [math]p_Y[/math] given by
- Find the ordinary generating functions [math]h_X(z)[/math] and [math]h_Y(z)[/math] for these distributions.
- Find the ordinary generating function [math]h_Z(z)[/math] for the distribution [math]Z = X + Y[/math].
- Show that [math]h_Z(z)[/math] cannot ever have the form
[[math]] h_Z(z) = \frac{z^2 + z^3 +\cdots+ z^{12}}{11}\ . [[/math]]
Hint: [math]h_X[/math] and [math]h_Y[/math] must have at least one nonzero root, but [math]h_Z(z)[/math] in the form given has no nonzero real roots. It follows from this observation that there is no way to load two dice so that the probability that a given sum will turn up when they are tossed is the same for all sums (i.e., that all outcomes are equally likely).
Show that if
then
and