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function (pmf) is represented by f(x) = P[X=x], i.e., the probability that the
random variable X takes the value x.
The mass distribution function must satisfy the conditions that
f(x) >0, for all x,
and
A cumulative distribution functio
n (cdf) is defined as
Next, we will define a number of functions to calculate discrete probability
distributions. We suggest that you create a sub-directory, say,
HOME\STATS\DFUN (Discrete FUNctions) where we will define the probability
mass function and cumulative distribution function for the binomial and Poisson
distributions.
Binomial distribution
The probability mass function of the binomial distribution is given by
where (
n
x
) = C(n,x) is the combination of n elements taken x at a time. The
values n and p are the parameters of the distribution. The value n represents
the number of repetitions of an experiment or observation that can have one of
two outcomes, e.g., success and failure. If the random variable X represents the
number of successes in the n repetitions, then p represents the probability of
getting a success in any given repetition. The cumulative distribution function
for the binomial distribution is given by
0.1)( =
∑
xall
xf
∑
≤
=≤=
xk
kfxXPxF )(][)(
nxpp
x
n
xpnf
xnx
,...,2,1,0,)1(),,( =−⋅⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
−