Probability distributions Assignment Help - Normal, Binomial, Poisson, Hyper geometric

Statistics Assignment Help >> Probability distributions Assignment Help - Normal, Binomial, Poisson

Probability distributions Assignment Help - Normal, Binomial, Poisson, Hyper geometric

In the subject of probability distributions and statistics, binomial is termed as the discrete probability distributions of the number of successes which are in a sequence of N independent yes or no experiments each of these yield success with probability p. Bernoulli distribution and Bernoulli trial are the other names given to these success or failure experiments. The binomial distribution is termed as binomial distribution when n equals to 1. When this Bernoulli distribution is repeated n times, the Bernoulli trial is formed. The Bernoulli distribution forms the basis of the popular binomial test, which is of statistical significance.

The binomial distribution finds its use in the modeling of successes when a sample of size n is drawn with a replacement from a population of size N. When the sampling is carried out without doing replacements, then the draws we get are not independent and hence the resulting distribution is a hyper geometric distribution and not a binomial one. In cases when the value of N is much larger than n, the binomial distribution formed is considered a good approximation and is widely use.

Probability mass function-by this measure, we can get the probability of exactly k successes in n number of trials. This is when the random variable K follows the binomial distribution and has parameters n and p. the relationship can be denote as K ~ B(n, p) and,

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If the distribution function is cumulative, it can be expressed as

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Here is the ‘floor’ under x, and is. The greatest integer which is less than or equal to x

The Bernoulli distribution- it is formed when n=1, and is a special case of binomial distribution. In symbolic terms, the meaning of X ~ B (1, p) and X ~ Bem (p) is one and same. In other words, any Bernoulli distribution B (n, p) can be said to be a sum of n independent Bernoulli trials or Bem (p), each having the same probability p.

The Poisson binomial distribution- is a special case of distribution when the sum of n independent Bernoulli trials is Bem (pi)

Also, If X has the position distribution with p1 = … = pn =p then X ~ B (n, p).

Normal approximation-when the value of n is not very large then the distribution obtained is also not too great. If we us a suitable continuity correction, then an excellent approximation to B (n, p) can be obtained which can be given by the normal distribution

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The improvement in approximation is found when n increases to at least 20 and is even better when the value of p is not near to o or 1.for deciding whether n is large enough or if p is far enough form the extremes of 0 or 1 , rules of thumb are used.

The hyper geometric probability distribution- is a probability distribution which is used to describe the number of successes for a sequence of n draws finite population without any replacement.

For example, when a random variable X follows any hyper geometric distribution of parameter N, n and m , the probability mass function is given by the equation:-
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