Question

Fit a poisson distribution to the following data : X 0 1 2 3 4 F 46 38 22 9 1

Answer 1

Answer:

In order to fit the Poisson distribution, we must estimate a value for λ from the observed data. Since the average count in a 10-second interval was 8.392, we take this as an estimate of λ (recall that the E(X) = λ) and denote it by ˆλ.

Answer 2

Answer:

You can fit a Poisson distribution to the given data and estimate the parameter λ.

Step-by-step explanation:

From the above question,

They have given :

To fit a Poisson distribution to the given data, you need to estimate the parameter λ of the Poisson distribution. The parameter λ is the mean number of events per unit of time or space, and it can be estimated from the sample mean of the data.

Here is a step-by-step procedure to fit a Poisson distribution to the given data:

Calculate the sample mean: The sample mean μ is given by the sum of the data values divided by the number of data points:

          μ = (0 + 1 + 2 + 3 + 4) / 5

             = 10 / 5

         μ  = 2

Since the Poisson distribution has only one parameter, we can use the sample mean μ to estimate λ:

          λ = μ = 2

Calculate the likelihood: Given the estimated value of λ, we can calculate the likelihood of the data by taking the product of the Poisson probabilities for each data point:

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          L = P(X = 0)$.^{46}$ * P(X = 1)$.^{38}$ * P(X = 2)$.^{22}$ * P(X = 3)$.^{9}$ * P(X = 4)$.^{1}$

where P(X = k) is the Poisson probability mass function given by:

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          P(X = k) = (λ$.^{k}$* e-λ) / k!

Maximize the likelihood:

     To find the best-fitting Poisson distribution, you need to find the value of λ that maximizes the likelihood L. You can do this by numerical optimization methods, such as the Newton-Raphson method or the expectation-maximization (EM) algorithm.

Check the goodness of fit: Finally,

            You can check the goodness of fit by comparing the observed and expected frequencies, and by calculating a test statistic such as the chi-square statistic.

By following these steps,

            You can fit a Poisson distribution to the given data and estimate the parameter λ.

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