Question

A box contains 7 items, 4 of which are defective. a random sample of 3 items are taken from the box. Let X be the number of defective items in the sample. 1.Find the probability mass function of X. 2.Find the mean and the variance of X.

Answer #1

here X follows hypergeometric distribution with parameter n=3 ; k=4 and N =7

probability mass function of X =P(X=x)=P(getting x defective from 4 and 3-x good items from 3)

**P(X=x)= ^{4}C_{x}*^{3}C_{3-x}/^{7}C_{3}**

**from above formula**

x | P(x) |

0 | 0.0286 |

1 | 0.3429 |

2 | 0.5143 |

3 | 0.1143 |

2)

x | P(x) | xP(x) |
x^{2}P(x) |

0 | 0.0286 | 0.000 | 0.000 |

1 | 0.3429 | 0.343 | 0.343 |

2 | 0.5143 | 1.029 | 2.057 |

3 | 0.1143 | 0.343 | 1.029 |

total | 1.714 | 3.429 | |

E(x) =μ= | ΣxP(x) = | 1.7143 | |

E(x^{2}) = |
Σx^{2}P(x) = |
3.4286 | |

Var(x)=σ^{2} = |
E(x^{2})-(E(x))^{2}= |
0.490 |

from above mean =1.7143 (this can directly be derived from formula nk/N =3*4/7=1.7143)

and variance =0.490 ( this can directly be derived from formula nk/N(1-k/N)*(N-n)/(N-1))

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