Final answer:
The probability generating function of a random variable X is G(t) = E[t^X], an analytical function encoding the probabilities of X. For a Bernoulli distribution with parameter theta, the PGF is G(t) = (1-theta) + theta*t, representing the distribution of a variable with two possible outcomes.
Step-by-step explanation:
The probability generating function (PGF) of a random variable X is defined as the power series G(t) = E[t^X], where E denotes the expected value, and the powers of t correspond to the values that the random variable X can take. The PGF is a way of encoding the probabilities associated with the random variable into a single analytical function.
For a Bernoulli distribution with parameter theta, the random variable X can take two values: 0 (failure) with probability 1-theta, and 1 (success) with probability theta. The PGF for a Bernoulli distribution is obtained by considering the expected value of t raised to X:
G(t) = E[t^X] = P(X=0)*t^0 + P(X=1)*t^1 = (1-theta) + theta*t.
The PGF for a Bernoulli random variable facilitates computations of probabilities and can be used to derive properties of the distribution such as mean and variance.