Sum

Three numbers are selected at random (without replacement) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X.Also, find the mean and variance of the distribution.

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#### Solution

X = larger of of three numbers

X = 3, 4, 5, 6

`P(x=3)=6xx1/6xx1/5xx1/4=1/20`

P (x = 4) = `18xx1/6xx1/5xx1/4=3/20`

P (x = 5) = `36xx1/6xx1/5xx1/4=6/20`

P (x = 6) = `60xx1/6xx1/5xx1/4=10/20`

Xi | Pi | PiXi | PiXi^{2} |

3 |
`1/20` |
`3/20` |
`9/20` |

4 |
`3/20` |
`12/20` |
`48/20` |

5 |
`6/20` |
`30/20` |
1
`50/20` |

6 |
`10/20` |
`60/20` |
`360/20` |

`"Mean"=sumP_iX_i=(105/20)=5.25`

`sumP_iX_i^2=567/20`

Var(X)= `sumP_iX_i^2-(sumP_iX_i)^2`

`=567/20-(105/20)^2=0.787`

Concept: Mean of a Random Variable

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