Stats 101

For Students

Solutions to Odd-Numbered End-of-Chapter Exercises

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Chapter 2

Review of Probability

2.1. (a) Probability distribution function for Y

Outcome (number of heads) | Y  0 | Y  1 | Y  2 |

Probability | 0.25 | 0.50 | 0.25 |

(b) Cumulative probability distribution function for Y

Outcome (number of heads) | Y  0 | 0 Y  1 | 1 Y  2 | Y 2 |

Probability | 0 | 0.25 | 0.75 | 1.0 |

(c) .

Using Key Concept 2.3:

and

so that

2.3. For the two new random variables and we have:

(a)

(b)

(c)

2.5. Let X denote temperature in F and Y denote temperature in C. Recall that Y  0 when X  32 and Y 100 when X  212; this implies Using Key Concept 2.3, X  70oF implies that and X  7oF implies

2.7. Using obvious notation, thus and This implies

(a) per year.

(b) , so that Thus where the units are squared thousands of dollars per year.

(c) so that and thousand dollars per year.

(d) First you need to look up the current Euro/dollar exchange rate in the Wall Street Journal, the Federal Reserve web page, or other financial data outlet. Suppose that this exchange rate is e (say e  0.80 Euros per dollar); each 1 dollar is therefore with e Euros. The mean is therefore

e C (in units of thousands of Euros per year), and the standard deviation is e C (in units of thousands of Euros per year). The correlation is unit-free, and is unchanged.

2.9. | | Value of Y | Probability Distribution of X |

| | 14 | 22 | 30 | 40 | 65 | |

| Value of X | 1 | 0.02 | 0.05 | 0.10 | 0.03 | 0.01 | 0.21 |

| | 5 | 0.17 | 0.15 | 0.05 | 0.02 | 0.01 | 0.40 |

| | 8 | 0.02 | 0.03 | 0.15 | 0.10 | 0.09 | 0.39 |

| Probability distribution of Y | 0.21 | 0.23 | 0.30 | 0.15 | 0.11 | 1.00 |

(a) The probability distribution is given in the table above.

(b) The conditional probability of Y|X  8 is given in the table below

Value of Y |

14 | 22 | 30 | 40 | 65 |

0.02/0.39 | 0.03/0.39 | 0.15/0.39 | 0.10/0.39 |...