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Poisson Distribution - Mathematical Methods of Physics, UGC - NET Physics | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

A Poisson distribution is the probability distribution that results from a Poisson experiment.

 

Attributes of a Poisson Experiment

Poisson experiment is a statistical experiment that has the following properties:

  • The experiment results in outcomes that can be classified as successes or failures.

  • The average number of successes (μ) that occurs in a specified region is known.

  • The probability that a success will occur is proportional to the size of the region.

  • The probability that a success will occur in an extremely small region is virtually zero.

Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.


Notation

The following notation is helpful, when we talk about the Poisson distribution.

  • e: A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.)

  • μ: The mean number of successes that occur in a specified region.

  • x: The actual number of successes that occur in a specified region.

  • P(x; μ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.

 

Poisson Distribution

Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution.

Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula:

Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e) (μx) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

The Poisson distribution has the following properties:

  • The mean of the distribution is equal to μ .

  • The variance is also equal to μ .

Poisson Distribution Example

The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?

Solution: This is a Poisson experiment in which we know the following:

  • μ = 2; since 2 homes are sold per day, on average.

  • x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.

  • e = 2.71828; since e is a constant equal to approximately 2.71828.

We plug these values into the Poisson formula as follows:

P(x; μ) = (e) (μx) / x!

P(3; 2) = (2.71828-2) (23) / 3!

P(3; 2) = (0.13534) (8) / 6

P(3; 2) = 0.180

Thus, the probability of selling 3 homes tomorrow is 0.180 .

 

Cumulative Poisson Probability

cumulative Poisson probability refers to the probability that the Poisson random variable is greater than some specified lower limit and less than some specified upper limit.

Cumulative Poisson Example

Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?

Solution: This is a Poisson experiment in which we know the following:

  • μ = 5; since 5 lions are seen per safari, on average.

  • x = 0, 1, 2, or 3; since we want to find the likelihood that tourists will see fewer than 4 lions; that is, we want the probability that they will see 0, 1, 2, or 3 lions.

  • e = 2.71828; since e is a constant equal to approximately 2.71828.

To solve this problem, we need to find the probability that tourists will see 0, 1, 2, or 3 lions. Thus, we need to calculate the sum of four probabilities: P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5). To compute this sum, we use the Poisson formula:

P(x < 3, 5) = P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5)

P(x < 3, 5) = [ (e-5)(50) / 0! ] + [ (e-5)(51) / 1! ] + [ (e-5)(52) / 2! ] + [ (e-5)(53) / 3! ]

P(x < 3, 5) = [ (0.006738)(1) / 1 ] + [ (0.006738)(5) / 1 ] + [ (0.006738)(25) / 2 ] + [ (0.006738)(125) / 6 ]

P(x < 3, 5) = [ 0.0067 ] + [ 0.03369 ] + [ 0.084224 ] + [ 0.140375 ]

P(x < 3, 5) = 0.2650

Thus, the probability of seeing at no more than 3 lions is 0.2650.

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FAQs on Poisson Distribution - Mathematical Methods of Physics, UGC - NET Physics - Physics for IIT JAM, UGC - NET, CSIR NET

1. What is the Poisson distribution?
Ans. The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, where these events occur with a known average rate and independently of the time since the last event.
2. How is the Poisson distribution used in mathematical methods of physics?
Ans. In mathematical methods of physics, the Poisson distribution is used to model various physical phenomena that involve the occurrence of random events. It is particularly useful in areas such as quantum mechanics, statistical mechanics, and particle physics, where the discrete nature of events needs to be taken into account.
3. What are the key properties of the Poisson distribution?
Ans. The key properties of the Poisson distribution are: - The mean and variance of the distribution are equal, given by λ (the average rate of events). - The distribution is discrete and defined for non-negative integer values. - The probability of observing a certain number of events in a fixed interval is independent of the length of the interval. - The events occur independently of each other. - The distribution is memoryless, meaning that the probability of an event occurring in the future is not affected by the past.
4. How can the Poisson distribution be applied to UGC - NET Physics exam questions?
Ans. In UGC - NET Physics exams, the Poisson distribution can be applied to various statistical and probability-related questions. For example, it can be used to calculate the probability of a certain number of particles decaying within a given time interval, the probability of a certain number of photons being detected in a given time interval, or the probability of a certain number of events occurring in a physical system.
5. What are some real-world examples where the Poisson distribution is applicable in physics?
Ans. The Poisson distribution finds applications in various real-world scenarios in physics, such as: - Modeling the arrival of individual particles or photons in a detector. - Analyzing the decay of radioactive nuclei. - Predicting the number of cosmic ray events in a certain area. - Estimating the number of photons emitted by a source in a given time period. - Studying the distribution of particles in a gas or liquid system.
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