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Random Variable and Probability Distributions
A variable that takes different probability in a random way so that the outcomes are not known is called a random variable. Hence, X is called a random variable if it takes values X_{1 }, X_{2 , }X_{3}. . .X_{n }randomly. There are several types of probability distribution such as the binomial distribution, Poisson, Normal etc. They differ according to the nature of the random variable.
Bernoulli Process:
An experiment consisting only two outcomes is known as “Bernoulli Process”. Generally, the two outcomes are called success and failure. For example, tossing a coin, result of an examination, throwing a dice, production of bulbs etc.
Binomial Distribution:
The discrete probability distribution derived from the Bernoulli process is known as Binomial distribution. The following are the basic assumptions under which Binomial Distribution can be used:
When the probability of a success in one trial is known, the probabilities of success of exactly once, twice, thrice, . . . etc. in n trials can also be known.
Let a trial be repeated so as to make a set of n trials. We denote the occurrence of an event known as success by S and the non-occurrence, a failure by F. Let p and q be the probabilities of a success and a failure in one trial respectively be such that p + q = 1. Let us assume that the trials are independent and the probability of success in every trial is the same. Now, we find the probabilities of 0, 1, 2 , . . . .,n success in n trials. Thr probabilities of a success and a failure be denoted by P(S) and P(F) respectively. Then the probability of r success and (n-r) failure in a set of n trials in any specified order say S.S.S.S …S (r times) F.F.F.F.F.F…F.(n-r times).
=P(S).P(S)… P(S) P(F).P(F).P(F)….. p(F)
=(p.p.p……r times) (q.q.q.q…. (n-r)times)
= p^{r} . q^{n-r}
But the number of orders of occurring r success out of n trials is same as the number of combination of n things taken r at a time i.e. ^{n}C_{r}. These are all equally probable and mutually exclusive hence by the theorem of total probability of r success; i.e. P(r) in a set of n independent trials is given by:
$$\;P(r)\;$$=^nC_r\;p^r\;q^_{n-r}\;\;$$……\;(0\leq\;r\leq\;n)\;$$
Thus if n = number of trials performed
p = probability of success in a trial
q = probability of a failure in a trial such that p + q = 1
r = number of success \;in trials
then P(r) = P(x = r) = probability of r success in n trials
$$\;=\;^nC_r\;p^r\;q^_{n-r}\;$$
The probabilities of 0, 1, 2, 3, . . . . . . n successes obtained by putting r = 0, 1, 2, 3, . . . . n in the above equation is listed below:
No of successes (r) | Probability of r success i.e. P(r) |
0 | P(0) = q^{n} |
1 | P(1) = C(n,1)p^{1}q^{n-1} |
2 | P(2) = C(n,2) P^{2}q^{n-2} |
3 | P(3) = C(n,3) P^{3}q^{n-3} |
n | P(2) = C(n,n) P^{n}q^{n-n} = p^{n} |
Mean and Standard deviation of Binomial Distribution:
If p be the probability of a success and q that of a failure in one trial, then the probabilities of 0,1,2,3, . . . . n successes in n trials are listed above which are the successive terms of the binomial expansion of (p+q)^{n }. Hence the distribution is known as binomial distribution. The mean and the standard deviation of the binomial distribution and np and (npq)^{-1/2} respectively.the two independent constants n and p (or q) are known as parameters.
Taken reference from
( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )
P(r) = P(x = r) = probability of r success in n trials
$$\;=^nC_r\;p^r\;q^_(n-r)\;$$
Mean of the distribution is given by np
Variance of the distribution is given by npq
$$Standard\;deviation\;of\;the\;distribution\;is\given\;by\;\sqrt{npq}\;$$
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