Embedded Files
S5 Binomial Random Variables
Among the most important random variable of elementary probability are the binomial random variables. These come from Bernoulli sequences by adding the number of 1's in the sequence. Here is the definition:
DEFINITION: Let B1, B2,...,BN be an IID sequence of N Bernoulli Random variables such that P(B(i)=1)=p for all i. Let X=B1+B2+...+BN be the sum of all N Bernoulli RV's. Then we say that X is Binomial with N Trials and success probability p. This is written as Bi(N,p).
The above provides a constructive definition, showing how a Binomial Random Variable is built from the sum of independent Bernoullis with the same success probability. An alternative method for defining random variables is to name their possible outcomes and attach probabilities to each outcome. For the Bi(N,p) random variable X, this can be done as follows:
X is Bi(N,p), if the set of possible outcome of X ranges from 0,1,..,N and the probabilities of the outcomes are:
We have already seen that this formula gives us the probability of exactly K 1's in a Bernoulli sequence. Thus the probability that sum of the Bernoullis X equals K is exactly equal to this probability. It is worth noting that X can range from 0 to N -- 0 being the case when there are no 1's so the entire sequence consists of 0's. The other extreme is N where the entire sequence consists of 1's
Page updated
Google Sites
Report abuse