Write the properties of Poisson distribution. Fit a poison distribution and find the expected frequencies.

X 0 1 2 3 4 5 6 7
Y 71 112 117 57 27 11 3 1

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A descrete random variable ‘x’ is said to follow possion distribution if its probability mass function is given by \(P(X=x) = \frac{e^{-λ} λ^{x}}{x!}\)

This can be written as X ~ P(X) and read as X follows possion distribution with parameter x.

Properties:

  1. It is a descrite probability distribution because the random variable x takes values 0,1,2,3,4,5 ……. ∞
  2. The mean and varience of possion distribution is λ

Problem Part:

x f fx
0 71 0
1 112 112
2 117 234
3 57 171
4 27 108
5 11 55
6 3 18
7 1 7
N = 399 Σfx = 705

n = 7 (becase n takes values upto 7)

Now,

\(\bar{x} = \frac{\sum fx}{N}\) = \(\frac{705}{399}\) = 1.77

∴ λ = 1.77

Calculation of Expected frequency:

we know, P(X=x) = P(x) = \(\frac{e^{-λ} λ^{x}}{x!}\)

Also, Expected frequency (fe) = N . P(x)

X \(P(X=x) = \frac{e^{-λ} λ^{x}}{x!}\) fe = P(X=x) . N
0 P(X=0) = \(\frac{e^{-1.77} 1.77^{0}}{0!}\) = 0.17 0.17 × 399 = 67.83 ~ 67
1 P(X=1) = 0.302 0.302 × 399 = 120.498 ~ 120
2 P(X=2) = 0.267 0.267 × 399 = 106.533 ~ 106
3 P(X=3) = 0.157 0.157 × 399 = 62.643 ~ 63
4 P(X=4) = 0.07 0.07 × 399 = 27.93 ~ 28
5 P(X = 5) = 0.025 0.025 × 399 = 9.975 ~ 10
6 P(X = 6) = 0.07 0.007 × 399 = 2.793 ~ 3
7 P(X = 7) = 0.002 0.002 × 399 = 0.798 ~ 1
ΣP(X=x) = 1 Σfe = 399

Here, We see the possion distribution is fitted and fitted posion distribution is

x 0 1 2 3 4 5 6 7
fe 68 120 106 63 28 10 3 1
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