Binomial Distribution

Last week we discussed Random Variable - mean and variance. Continuing
on statistics, this week we are going to discuss Binomial Distribution.

To
begin Binomial distribution, we must know that a lot of experiments are
possible where there are only two outcomes (or we assume that any other
outcome is ruled out). For example when we toss a coin the only
possible outcomes are either a Head or a Tail (and we dont assume that
the coin will fall vertically and remain upright, it will fall flat
eventually).  So, one of them can be termed as
singlequotesuccesssinglequote and another as
singlequotefailuresinglequote. Such trials which have only two outcomes
are called Bernoulli Trial after Swiss scientist Jacob
Bernoulli. In Bernoulli trial the outcome of each trial in independent
of previous trials, the probability of success and failiue remains same
in each trial.

For example, if we toss a coin then there
are only two possible outcomes and the probability of a head or tail
remains constant in each trial which is independent of the outcome of
previous trial.

Binomial Distribution: Binomial
distribution is a discrete probability distribution of a sequence of n
consecutive experiments (or trials) where each trial has a chance of
success of p and chance of failure is q (where p + q = 1).

For example
we a toss a coin three times (here n = 3 ) and we want to know the
probability of heads two times. In other words, if outcome HEAD is a
success we are looking for two successes. The possible favorable
outcomes will be HTH,HHT,THH i.e., chosing 2 places out of 3. It is
possible in ³C₂ ways.

So,
P(HTH) = p.q.p = p²q
P(HHT) = p.p.q = p²q
P(THH) = q.p.p = p²q

Total probability = p²q + p²q + p²q = 3p²q = ³C₂p²q

P(X = 2) = ³C₂p²q^3-2
The above expression can be genralized as:

P(X = r) = ³C_rp^rq^3-r  ( n = 3)

So, we get

If the number of trials is n the generalized formula becomes:
P(X = r) =  ⁿC_rp^rq^n-r

Example:
Five dice are thrown simultaneoudsly. If the occurence of an even
number in a single throw is considered a success, find the probability
of at most 3 successes.

Solution: Probability of at most
three success = probabilty of one success + probabilty of one successes
+ probabilty of three successes.
Here probabiliry of success, p = 3/6 = 1/2
Probability of failure,


              = P(X=0) + P(X=1) + P(X=2) + P(X=3)
              = ⁵C₀p⁰q⁵ + ⁵C₁p¹q⁴ + ⁵C₂p²q³ + ⁵C₃p³q²


Mean and Variance of Binomial Distribution:

For a Binomial Variate X with success probability p and number of experiment n is given by:


Variance: