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## Distributions

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**Raw data**Frequency distribution No.of samples fall in the interval Classes Class-interval midpoints (Largest value - smalest value) (frequency) (Difference of this / desirable classes) Relative frequency Cumulative frequency**The binomial distribution gives the discrete**probability distribution of obtaining exactely n success out of N Bernoulli trials (where the result of each Bernoulli trials is true with probability p and false with probability q =1- p). The binomial distribution is therefore given by Binomial Distribution where is a binomial coefficient.**Binomial Distribution**• If the probability of each outcome remains the same throughout the trials , then such trials is called Bernoulli trials and the experiment having n Bernoulli trials is called a binomial experiment. • Example Let X have a binomial distribution with n = 4 and p = 1/3. find P(X = 1), P( X = 3/2), P( X = 6) and P(X ≤2).**The binomial probability distribution for**n = 4 and p = 1/3, is for x = 0,1,2,3,4 ; because a r.v X with a binomial distribution takes only one of the integer values 0,1,2…n.**Poisson Distribution**• Poisson is the name of French mathematician Sime’on Denis Poisson (1781-1840) and it is published in 1837. • It is a limiting approximation of the binomial distribution b (x; n, p). • If we assume that n goes to infinity and p approaches to zero in such a way that = np remains constant, then the limiting form of the binomial probability distribution is**x = 0,1,2,…,**where The Poisson distribution has only one parameter >0, and is denoted by p (x; ). The Poisson probability distribution is also called the law of small number or the rare events distribution. It has a wide application in the field of Physics, Biology , Operation Research and Management Sciences.**It is appropriate when the number of possible occurrences is**very large but the number of actual occurrences is very small in a fixed period of time. Example : Fit a Poisson distribution to these data And compute the theoretical frequencies. which is an estimate of **The probabilities are computed by using the Poisson**recurrence formula. for x = 1,2,3,….**In case the table values for are not available,**they are computed by use of logarithms**A normal distribution in a variate X with mean and**variance ² has probability function P(x) = 1.e-(x-)²/(2s²)/2 on the domain x ∈(∞‚ ∞)this type of distribution is mostely used in the theory of errors. Due to its curved flaring shape social scientists refer to it as the “bell curve” and physicist generally called it Gaussian distribution. Feller(1968) uses the symbols x for P(x) in the above equation, but then swithes to n(x) in Feller(1971). The so called "standard normal distribution" is given by taking = 0 and ² = 1 in a general normal distribution. A normal distribution is the limiting case of discrete binomial distribution. ∞ Normal Distribution N ∞, Np = Binomial Poisson ∞ N Normal/Gaussian