**2. Two fair six-sided dice are tossed independently. Let M = the maximum of the two…**

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## Question “2. Two fair six-sided dice are tossed independently. Let M = the maximum of the two…”

## Answer

**Solution:**

we are given that : two fair six sided dice are

tossed independently.

M = Maximum of the two tosses.

Thus we get:

First Die | Second Die | M = Maximum of two dice |

1 | 1 | 1 |

1 | 2 | 2 |

1 | 3 | 3 |

1 | 4 | 4 |

1 | 5 | 5 |

1 | 6 | 6 |

2 | 1 | 2 |

2 | 2 | 2 |

2 | 3 | 3 |

2 | 4 | 4 |

2 | 5 | 5 |

2 | 6 | 6 |

3 | 1 | 3 |

3 | 2 | 3 |

3 | 3 | 3 |

3 | 4 | 4 |

3 | 5 | 5 |

3 | 6 | 6 |

4 | 1 | 4 |

4 | 2 | 4 |

4 | 3 | 4 |

4 | 4 | 4 |

4 | 5 | 5 |

4 | 6 | 6 |

5 | 1 | 5 |

5 | 2 | 5 |

5 | 3 | 5 |

5 | 4 | 5 |

5 | 5 | 5 |

5 | 6 | 6 |

6 | 1 | 6 |

6 | 2 | 6 |

6 | 3 | 6 |

6 | 4 | 6 |

6 | 5 | 6 |

6 | 6 | 6 |

**Part a) Probability mass function of M.**

We need to find frequencies of each possible values of M and

then divide each frequency by N = 36

M | f = frequency | P(M) |

1 | 1 | 0.0278 |

2 | 3 | 0.0833 |

3 | 5 | 0.1389 |

4 | 7 | 0.1944 |

5 | 9 | 0.2500 |

6 | 11 | 0.3056 |

N = 36 |

**Part b) Cumulative distribution function of M and Graph
it.**

To get cumulative distribution function , we need to find F(M)

column

That is find cumulative sum of probabilities of each value of

M.

Thus we get :

M | P(M) | Calculations of F(M) | F(M) |

1 | 0.0278 |
0.0278 | 0.0278 |

2 | 0.0833 | 0.0278+0.0833=0.1111 | 0.1111 |

3 | 0.1389 | 0.1111+0.1389=0.2500 | 0.2500 |

4 | 0.1944 | 0.2500+0.1944=0.4444 | 0.4444 |

5 | 0.2500 | 0.4444+0.2500=0.6944 | 0.6944 |

6 | 0.3056 | 0.6944+0.3056=1.0000 | 1.0000 |

**Part c) Expected value of M**

M | P(M) | M * P(M) |

1 | 0.0278 | 0.0278 |

2 | 0.0833 | 0.1667 |

3 | 0.1389 | 0.4167 |

4 | 0.1944 | 0.7778 |

5 | 0.2500 | 1.2500 |

6 | 0.3056 | 1.8333 |

Thus

**Part d) Variance of M**

Where

M | P(M) | M * P(M) | M^2 * P(M) |

1 | 0.0278 | 0.0278 | 0.0278 |

2 | 0.0833 | 0.1667 | 0.3333 |

3 | 0.1389 | 0.4167 | 1.2500 |

4 | 0.1944 | 0.7778 | 3.1111 |

5 | 0.2500 | 1.2500 | 6.2500 |

6 | 0.3056 | 1.8333 | 11.0000 |

**Part e) Standard deviation of M**

## Conclusion

Above is the solution for “**2. Two fair six-sided dice are tossed independently. Let M = the maximum of the two…**“. We hope that you find a good answer and gain the knowledge about this topic of **math**.