X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Y | 71 | 112 | 117 | 57 | 27 | 11 | 3 | 1 |

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Login NowA 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:

- It is a descrite probability distribution because the random variable x takes values 0,1,2,3,4,5 ……. ∞
- 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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