#### Question

Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

#### Solution

First six positive integers are {1, 2, 3, 4, 5, 6}

No. of ways of selecting 2 numbers from 6 numbers without replacement = ^{6}C_{2} = 15

X denotes the larger of the two numbers, so X can take the values 2, 3, 4, 5, 6.

Probability distribution of X:

X | 2 | 3 | 4 | 5 | 6 |

P(x) | 1/15 | 2/15 | 3/15 | 4/15 | 5/15 |

Computation of Mean and Variance:

X_{i} |
P(X=x_{i}) |
p_{i}x_{i} |
p_{i}x_{i}^{2} |

2 | 1/15 | 2/15 | 4/15 |

3 | 2/15 | 6/15 | 18/15 |

4 | 3/15 | 12/15 | 48/15 |

5 | 4/15 | 20/15 | 100/15 |

6 | 5/15 | 30/15 | 180/15 |

`sum p_ix_1=70/15=14/3` |
`sump_ix_^2=350/15=70/3` |

Mean `=sump_ix_i=70/15=4.67`

Variance `=sump_ix_i^2-(sump_ix_i)^2=70/3-196/9=(210-196)/9=14/9`

Is there an error in this question or solution?

Solution Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution. Concept: Mean of a Random Variable.