Question

Suppose that there are 5 dollar bills in a box: three 1 dollar bills, one 5 dollar bill and one 10 dollar bill. You are allowed to pick up two bills at the same time from the box randomly. Let X denote the money you get from this game. (a) What’s the support and p.m.f. of X? (b) What’s the mean and variance of X?

Answer #1

Let X is a random variable shows the amount you win excluding you pay to play. Here X can take values $2, $6 and $11,$15.

Total number of bills: 3+1+1=5

Number of ways of selecting 2 bills out of 5 is C(5,2) = 10

When X=$2, then both $1 bills are selected.

So

P(X=$2) = C(3,2) / 10 = 3/10 =0.30

When X=$6, then one $5 bill and one $1 bill is selected.

So

P(X=$6) = [C(3,1)*C(1,1)] / 10 = 0.30

When X=$11, then one $10 bill and one $1 bill is selected.

So

P(X=$10) = [C(3,1)*C(1,1)] / 10 = 0.30

When X=$15, then one $5 bill and one $10 bill is selected.

So

P(X=$15) = [C(1,1)*C(1,1)] / 10 = 0.10

The pmf of X is:

X | P(X=x) |

2 | 0.3 |

6 | 0.3 |

11 | 0.3 |

15 | 0.1 |

(b)

Following table shows the calculations for mean and variance:

X | P(X=x) | xP(X=x) | x^2P(X=x) |

2 | 0.3 | 0.6 | 1.2 |

6 | 0.3 | 1.8 | 10.8 |

11 | 0.3 | 3.3 | 36.3 |

15 | 0.1 | 1.5 | 22.5 |

Total | 7.2 | 70.8 |

The mean is:

The variance is:

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