Given the pdf of a Bernoulli distribution, and the value of x ranging from zero to n,determine it's mgf,mean and variance?

The probability mass function (pmf) of a Bernoulli distribution is given by:

P(X = x) = p^x * (1-p)^(1-x)

where p is the probability of success and x is the outcome (0 or 1).

The moment generating function (mgf) of a Bernoulli distribution is given by:

M(t) = E[e^(tX)] = p*e^t + (1-p)

The mean of a Bernoulli distribution is given by:

E[X] = p

The variance of a Bernoulli distribution is given by:

Var(X) = p*(1-p)

Therefore, for a Bernoulli distribution with x ranging from zero to n, the mgf, mean, and variance are as follows:

M(t) = p*e^t + (1-p)
E[X] = p
Var(X) = p*(1-p)