Question

Three tables listed below show random variables and their probabilities. However, only one of these is actually a probability distribution. A B C x P(x) x P(x) x P(x) 25 0.3 25 0.3 25 0.3 50 0.1 50 0.1 50 0.1 75 0.2 75 0.2 75 0.2 100 0.4 100 0.6 100 0.8 a. Which of the above tables is a probability distribution? b. Using the correct probability distribution, find the probability that x is: (Round the final answers to 1 decimal place.) 1. Exactly 100 = 2. No more than 75 = 3. More than 75 = c. Compute the mean, variance, and standard deviation of this distribution. (Round the final answers to 2 decimal places.) 1. Mean µ 2. Variance σ2 3. Standard deviation σ

Answer #1

(a)

From the given data, the following Table is formed:

For A:

x | P(x) |

25 | 0.3 |

50 | 0.1 |

75 | 0.2 |

100 | 0.4 |

Total = | 1.0 |

For B:

x | P(x) |

25 | 0.3 |

50 | 0.1 |

75 | 0.2 |

100 | 0.6 |

Total = | 1.2 |

For C;

x | P(x) |

25 | 0.3 |

50 | 0.1 |

75 | 0.2 |

100 | 0.8 |

Total = | 1.4 |

It is noted that in case of A only, Total Probability = 1.

So,

Only A is actually a probability distribution.

(b)

(1)

P(x = 100) = **0.4**

(2)

P(X75) = P(X=25) +
P(X=50) + P(X=75) = 0.3 + 0.1 + 0.2 = **0.6**

(3)

P(X>75) = **0.4**

(c)

From the Probability Distribution, the following Table is formed:

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

25 | 0.3 | 7.5 | 187.5 |

50 | 0.1 | 5 | 250 |

75 | 0.2 | 15 | 1125 |

100 | 0.4 | 40 | 4000 |

Total = | 1.0 | 67.50 | 5562.50 |

(1) Mean = **67.50**

(2) Variance = E(X^{2}) - (E(X))^{2} = 5562.50 -
67.50^{3} = **1006.25**

(3) Standard Deviation =

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