Types of Probablity distributions
If our actually observed data do not match the data expected on the basis of assumptions, we would have serious doubts about our assumptions. Such data of assumption often lead to theoretical frequency distributions also known as probability distribution. This distribution is not based on actual experimental data but on certain theoretical considerations. This may be simple two valued distribution like 3:1 as in Mendelian cross or it may be more complicated. Some of the most important probability distributions are,
- Gaussian/Normal distribution
- Binomial distribution
- Poisson distribution
Binomial and Poisson distribution apply to the discontinuous random variables and are together known as discontinuous distributions. Normal distribution applies to continuous random variables and is called as continuous distribution.
French Mathematician-cum-Physicist Simeon Denis Poisson discovered Poisson distribution in 1837. It is also known as discrete distribution. This was discovered as a limiting event for Binomial distribution. For n-trials the binomial distribution is (q + p) n; the probability of x successes is given by P(X=x) = nCx p x qn-x. If the number of trials “n” is very large and the probability of success ‘p’ is very small then the product np = m, is non-negative and finite.
Poisson distribution is widely used in cases where chance of any individual event being success is small and the number of trials tends to be infinite. This distribution is used to describe the rare events. Some examples of Poisson distribution are:
- The number of blinds born in a town in a particular year.
- Number of mistakes committed in a typed page.
- The number of students scoring very high marks in all subjects
- The number of plane accidents in a particular week.
- The number of defective screws in a box of 100, manufactured by a reputed company.
- Number of suicides reported in a particular day.
Definition of Poisson distribution
The discontinuous random variable x is said to follow Poisson distribution if it assumes only non-negative values and its probability density function is given by,
Here m is known as parameter of the distribution so that m >0. Since number of trials is very large and the probability of success p is very small, it is clear that the event is a rare event. Therefore Poisson distribution relates to rare events.
Characteristics of Poisson distribution
- Discrete distribution: Poisson distribution is a discrete distribution like Binomial distribution, where the random variable assumes as a countably infinite number of values 0, 1, 2 ….
- Applied when rate of success is very small and rate of failure is very high: Poisson distribution is applied in situations where the probability of success p of an event is very small and that of failure q is very high. Also here, “n” is very large.
- Parameter of Poisson distribution: The main parameter of the Poisson distribution is m. If the value of m is known, all the probabilities of the Poisson distribution can be determined.
- Mode of distribution: Mean = m = variance; so that standard deviation = m. Poisson distribution may have one mode or two modes of distribution.
- Additive Property: If X and Y are two independent Poisson distributions with parameter m1 and m2 respectively. Then (X+Y) also follows the Poisson distribution with parameter (m1 + m2)
- As an approximation to binomial distribution: Poisson distribution can be taken as a limiting form of Binomial distribution when n is large and p is very small. Here the product np=m which remains constant.
- Assumptions: The Poisson distribution is based on the following assumptions.
*The occurrence or non- occurrence of an event does not influence the occurrence or non-occurrence of any other event.
*The probability of success for a short time interval or a small region of space is proportional to the length of the time interval or space as the case may be.
*The probability of the happening of more than one event is a very small interval is negligible.
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