#### 9. Populations and Probabilities

*Question 1*

(a) This is a valid probability model since all probabilities are nonnegative and sum to 1.

(b) This is not a valid probability model since P(C) is negative, even though the total is 1.

(c) This is not a valid probability model since the sum of the probabilities is 1.6, not 1.

(d) This is not quite a valid probability model since the sum of the probabilities is 0.99, not 1. To accurately assign equal probabilities fractions (1/3) should be used.

*Question 2*

The surprisal is -log_{2}(*p*) = 2.5. Rearranging gives *p* = 2^{-2.5} = **0.1768**.

*Question 3*

From Section 9.5 we know the height of the uniform density is 1/40. The probability to the right of a particular value *x* is

P(*X* ≥ *x*) = base × height = (190 - *x*) × (1/40).

We want this to be *p*. Solving gives 40*p* = 190 - *x*, so ** x = 190 - 40p**.

For example, how tall would you be if you were in the top 10% of heights in this distribution? Substituting *p* = 0.1 into the formula gives *x* = 190 - 40(0.1) = 186, so the top 10% of heights consists of females taller than 186 cm.

*Question 4*

This question is much harder than the previous one. From Section 9.5 we know the height of the triangular density is 1/20. If *x* is 170 or more then the height of the density curve at *x* is (1/20)×((190 - *x*)/20). You can check that this gives height 0 for *x*=190 and height 1/20 for *x*=170. The area to the right of *x* is then

P(*X* ≥ *x*) = (1/2) × base × height = (1/2) × (190 - *x*) × (1/20)×((190 - *x*)/20) = (1/800)×(190 - *x*)^{2}.

We want this to be *p*. Solving gives 800*p* = (190 - *x*)^{2}, so ** x = 190 - √(800p)**.

For example, how tall would you be if you were in the top 10% of heights in this distribution? Substituting *p* = 0.1 into the formula gives *x* = 190 - √(80) = 181.1, so the top 10% of heights consists of females taller than 181.1 cm.

This formula only works for *x* ≥ 170 (*p* ≤ 0.5). For *x* ≤ 170 (*p* ≥ 0.5) the area to the right is

P(*X* ≥ *x*) = 1 - (1/800)×(*x* - 150)^{2}.

Solving this to get *x* in terms of *p* gives ** x = 150 + √(800(1 - p))**.