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Characteristic Functions
Examples
1. Bernoulli Distribution
he Bernoulli distribution is a discrete distribution having two possible outcomes
labelled by n=0 and n=1 in which n=1 ("success") occurs with probability p
and ("failure") occurs with probability q=1-p, where 0 < p < 1. It therefore has
probability function
which can also be written
The corresponding distribution function is
The characteristic function is
Characteristic Function
• In probability theory and statistics, the binomial
distribution is the discrete probability
distribution of the number of successes in a
sequence of n independent yes/no experiments,
each of which yields success with probability p.
Such a success/failure experiment is also called
a Bernoulli experiment or Bernoulli trial. In fact,
when n = 1, then the binomial distribution is the
Bernoulli distribution. The binomial distribution is
the basis for the popular binomial test of
statistical significance
Example
A typical example is the following: assume 5% of the population is green-eyed. You pick
500 people randomly. The number of green-eyed people you pick is a random variable X
which follows a binomial distribution with n = 500 and p = 0.05 (when picking the people
with replacement
Probability mass function
In general, if the random variable X follows the binomial distribution with parameters n
and p, we write X ~ B(n, p). The probability of getting exactly k successes is given by
the probability mass function:
for k=0,1,2,...,n and where
Parameters
number of trials (integer)
success probability (real)
Support
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess Kurtosis
Entropy
mgf
Char. func.
one of