A binomial distribution has a probability of success = 0.8. Calculate the probability of exactly 3 successes in 8 trials:

f(k;n,p) = | n! * p^{k}q^{n - k} | |

k!(n - k)! |

q = 1 - p (q represents the probability of failure)

q = 1 - 0.8

q = 0.2

n! = 8!

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

8! = 40320

k! = 3!

3! = 3 * 2 * 1

3! = 6

(n - k)! = (8 - 3)!

(n - k)! = 5!

5! = 5 * 4 * 3 * 2 * 1

5! = 120

P(X = 3) = | 8! * 0.8^{3}0.2^{(8 - 3)} | |

3!(8 - 3)! |

P(X = 3) = | 40320 * 0.512 * 0.2^{5} | |

6 * 120 |

P(X = 3) = | 40320 * 0.512 * 0.00032 | |

720 |

P(X = 3) = | 6.6060288 | |

720 |

P(X = 3) =

μ = np

μ = 8 x 0.8

μ =

Since q = 1 - p, we have n(1 - p) = 8(1 - 0.8)

nq = 8(0.2)

nq = 1.6

σ

σ

σ

σ

σ = √σ

σ = √1.28

σ =

Skewness = | 1 - 2p |

√np(1 - p) |

Skewness = | 1 - 2(0.8) |

√8(0.8)(1 - 0.8) |

Skewness = | 1 - 1.6) |

√8(0.8)(0.2) |

Skewness = | -0.6 |

√1.28 |

Skewness =

Kurtosis = | 1 - 6p(1 - p) |

np(1 - p) |

Kurtosis = | 1 - 6(0.8)(1 - 0.8) |

8(0.8)(1 - 0.8) |

Kurtosis = | 1 - (4.8)(0.2) |

8(0.8)(0.2) |

Kurtosis = | 1 - 0.96 |

1.28 |

Kurtosis = | 0.04 |

1.28 |

Kurtosis =