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 ³ 2

Probability

0

0.25

0.75

1.0

(c) 

Using Key Concept 2.3:  and

so that

 

 

2.     We know from Table 2.2 that        So

(a)

(b)

 

 

(c)  Table 2.2 shows        So

 

6.     The table shows that              

(a)

(b)

(c)  Calculate the conditional probabilities first:

 

The conditional expectations are

 

(d)  Use the solution to part (b),

(e)  The probability that a randomly selected worker who is reported being unemployed is a college graduate is

The probability that this worker is a non-college graduate is

(f)  Educational achievement and employment status are not independent because they do not satisfy that, for all values of x and y,

For example,

 

 

10.   Using the fact that if  then  and Appendix Table 1, we have

(a)

(b)

 

(c)

(d)

 

12.   (a)  0.05

(b)  0.950

(c)  0.953

(d)  The tdf distribution and N(0, 1) are approximately the same when df is large.

(e)  0.10

(f)  0.01