Binominal distribution

Question

x 0 1 2 3

P(x) 0.4 0.3 0.2 0.1

Does x have a binominal distribution? Justify

Answer 1

The first thing to note is that the probabilities sum to 1, which means that there are no other outcomes.

There are 4 values so there have to be 4 binomial terms, implying (p+1-p)³=1, where p is a probability of success and q=1-p the probability of failure.

(p+q)³=p³+3p²q+3pq²+q³=0.4+0.3+0.2+0.1, where p is the probability of success and q the probability of failure. 0.4 is the probability of 3 successes; 0.3 that of 2 successes; 0.2 that of 1 success; 0.1 that of no successes. Therefore p³=0.4, p=0.74 approx, so q=0.26, so q³=0.02 approx, whereas the measured probability is 0.1. In fact the probability distribution is linear, not binomial.

If the distribution had been binomial we would have had (for example) p(0)=0.018, p(1)=0.153, p(2)=0.429, p(3)=0.4.