# 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

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