Probability generating functions: แบบฝึกหัด

The probability generating function of a discrete random variable \(X\) is given by \(G_X(t) = \frac{1}{3}(1 + t + t^2)\). A second independent random variable \(Y\) has the PGF \(G_Y(t) = \frac{1}{2}(1 + t)\). Find the probability \(P(X + Y = 2)\).

The PGF of the discrete random variable \(X\) is \(G_X(t) = \frac{1}{1-p(t-1)}\) where \(0 < p < 1\). Find the expression for the variance \(\text{Var}(X)\) in terms of \(p\).

The random variable \(X\) has a probability generating function given by \(G_X(t) = (0.2 + 0.8t)^n\). Given that the variance of \(X\) is 1.6, find the value of \(n\) and use it to find \(P(X=1)\).

The random variable \(X\) has PGF \(G_X(t) = \frac{k}{4-t-t^2}\). Find the value of the constant \(k\) and the mean \(E(X)\).

The PGF of a random variable \(X\) is given by \(G_X(t) = k(1-qt)^{-1}\) for \(0 < q < 1\). Given that \(G_X(1) = 1\), find the value of \(k\) and determine the expression for \(P(X=r)\) for \(r = 0, 1, 2, \).

The probability generating function of a random variable \(Y\) is given by \(G_Y(t) = k(2 + t + t^2)^2\), where \(k\) is a constant. Determine the value of \(k\) and find \(P(Y=1)\).

The independent random variables \(X\) and \(Y\) have probability generating functions \(G_X(t) = (0.7 + 0.3t)^5\) and \(G_Y(t) = (0.7 + 0.3t)^8\). Let \(Z = X + Y\). Find the probability \(P(Z=2)\) correct to 4 decimal places.

Let \(X\) be a random variable with probability generating function \(G_X(t)\). If \(G_X(t) = e^{2(t-1)}\), find the exact value of \(Var(X)\).

A discrete random variable \(X\) has a probability generating function given by \(G_X(t) = \frac{1}{(4-3t)^2}\).

(a) Find the value of \(P(X = 1)\).
(b) Use the properties of probability generating functions to find the mean \(E(X)\).
(c) Find the variance \(Var(X)\).
(d) A second independent random variable \(Y\) has the same distribution as \(X\). Find the probability generating function of \(Z = X + Y\) and hence state the value of \(E(Z)\).

A random variable \(Y\) is the sum of two independent discrete random variables \(X_1\) and \(X_2\). The pgf of \(X_1\) is \(G_{X_1}(t) = e^{2(t-1)}\) and the pgf of \(X_2\) is \(G_{X_2}(t) = (0.4 + 0.6t)^2\).

(a) Write down the pgf of \(Y\), \(G_Y(t)\).
(b) By differentiating \(G_Y(t)\), find the exact value of \(E(Y)\).
(c) Show that the variance of \(Y\) is the sum of the variances of \(X_1\) and \(X_2\).
(d) Find the coefficient of \(t^2\) in the expansion of \(G_Y(t)\) to determine \(P(Y=2)\), giving your answer to 4 decimal places.