The moment generating function (m.g.f.) of a random variable X is defined as M(t) = E(e^(tX)).

Given the probability density function (pdf) f(x) = qx^(p-1), x = 1,2,3,..., we can find the m.g.f. by taking the expected value of e^(tX).

M(t) = E(e^(tX)) = Σ [e^(tx) * f(x)] for all x in the domain of X.

Substituting the given pdf into the formula, we get:

M(t) = Σ [e^(tx) * qx^(p-1)] for x = 1,

Question

The moment generating function (m.g.f.) of a random variable X is defined as M(t) = E(e^(tX)).

Given the probability density function (pdf) f(x) = qx^(p-1), x = 1,2,3,..., we can find the m.g.f. by taking the expected value of e^(tX).

M(t) = E(e^(tX)) = Σ [e^(tx) * f(x)] for all x in the domain of X.

Substituting the given pdf into the formula, we get:

M(t) = Σ [e^(tx) * qx^(p-1)] for x = 1,

Knowee AI · Accepted Answer

The moment generating function (m.g.f.) of a random variable X is defined as M(t) = E(e^(tX)).

Given the probability density function (pdf) f(x) = qx^(p-1), x = 1,2,3,..., we can find the m.g.f. by taking the expected value of e^(tX).

M(t) = E(e^(tX)) = Σ [e^(tx) * f(x)] for all x in the domain of X.

Substituting the given pdf into the formula, we get:

M(t) = Σ [e^(tx) * qx^(p-1)] for x = 1,

Find the m g f of the pdf 𝑓(𝑥) = 𝑞𝑥−1𝑝, 𝑥 = 1,2,3, . . .. and hence evaluate its mean andvariance. (given p+q =1)