Probability generating functions (pgf)：練習問題

A discrete random variable \(X\) has a probability generating function (pgf) given by \(G_X(t) = 0.2 + 0.3t + 0.5t^2\). Find the probability that \(X = 1\).

Independent random variables \( X_1, X_2, \dots, X_n \) each follow a Bernoulli distribution with parameter \( p \). Let \( S = \sum_{i=1}^n X_i \). Use probability generating functions to determine the PGF of \( S \).

The probability generating function of a random variable \( X \) is given by \( G_X(t) = e^{2(t-1)} \). Let \( Y = 3X + 1 \). Find the probability generating function of \( Y \), \( G_Y(t) \).

The independent random variables \(X\) and \(Y\) have probability generating functions \(G_X(t)\) and \(G_Y(t)\) respectively. Let \(Z = X + Y\). Which of the following expressions represents the probability generating function of \(Z\)?

The probability generating function of a random variable \(X\) is given by \(G_X(t) = (0.6 + 0.4t)^3\). Find the value of \(P(X = 0)\).

The probability generating function of a discrete random variable \(X\) is \(G_X(t) = k(2 + t)^3\). First, show that \(k = \frac{1}{27}\), and then determine the variance \(Var(X)\).

The probability generating function for a random variable \(X\) is \(G_X(t)\). Show that \(G''_X(1) = E(X^2) - E(X)\) and use this result to derive the variance of a geometric distribution with parameter \(p\) in terms of \(p\) and \(q\), where \(q = 1-p\).

A discrete random variable \( X \) has the probability generating function (pgf) given by:
\( G_X(t) = k(2 + t + t^2)^2 \)

(a) Show that the constant \( k = \frac{1}{16} \).

(b) Find the mean \( E(X) \) and the variance \( Var(X) \) using the properties of pgfs.

(c) A second independent random variable \( Y \) has pgf \( G_Y(t) = \frac{1}{3}(1 + t + t^2) \). Find the probability \( P(X + Y = 1) \).