ABSTRACT
A binomial experiment involves n independent and identical trials such that each trial can result in to one of the two possible outcomes, namely, success or failure. If p is the probability of observing a success in each trial, then the number of successes X that can be observed out of these n trials is referred to as the binomial random variable with n trials and success probability p. The probability of observing k successes out of these n trials is given by the probability mass function
P (X = k|n, p) = ( n
k
) pk(1− p)n−k, k = 0, 1, ..., n. (3.1.1)
The cumulative distribution function of X is given by
P (X ≤ k|n, p) = k∑ i=0
( n
i
) pi(1− p)n−i, k = 0, 1, ..., n. (3.1.2)
Binomial distribution is often used to estimate or determine the proportion of individuals with a particular attribute in a large population. Suppose that a random sample of n units is drawn by sampling with replacement from a finite population or by sampling without replacement from a large population. The number of units that contain the attribute of interest in the sample follows a binomial distribution. The binomial distribution is not appropriate if the sample was drawn without replacement from a small finite population; in this situation the hypergeometric distribution in Chapter 4 should be used. For practical
of size
distribution with n trials and success probability p by is right-skewed when p < 0.5, and left-skewed when p = 0.5. See the plots of probability mass For large n, binomial distribution is approximately symmetric about its mean np.
