[;I + E and assume that these data are available: yl = 4, y2 = 3, y3 = 5, y 4 = 6 ; xl = 3, x2 = 2, x3 = 3 , x4 = 5

(1)

(Total 100 Points.) Tltere nre 5 Probkrris iri this a n n t Sltorv interr~iedinte steps nndforirurlm for pnrlinl credit. I'ou r?~ust a p l n k horv you coniprcte gorrr. results or ansivers forficll credit.

1. (20 points)

Let X and Y have the joint probability density distribution

(a) Find the marginal distribution o f X and 1'. (8 points) (b) Find P(Y > 5

I

X < 10). (6 points)

(c) Let U = X ^-Y, find the probability density fuilction of U. (6 points)

2. (20 points)

(a) Find the moment-generating fuilctioil of the random variable X having a chi-squared distribution with v degrees of freedom. (10 points)

(b) Use the moment-generating function in pait (a) to find the mean and variance of the chi-squared distribution with v degrees of freedom. (10 points)

3. (1 0 points)

It is claimed that a new diet will reduce a person's weight by 4.5 kilograms, on average, in a period of I0 days. The following data table recorded the weights of 10 men who followed this diet before and after a 10-day period.

Use the signed-rank test at the 0.05 level of significant to test the hypothesis that the diet reduces the Mail

WeightBefore

median weight by 4.5 kilograms against the alternative hypothesis that the median difference ill

1 2 3 4 5 6 7 8 9 1 0

58.5 60.3 61.7 69.0 64.0 62.6 56.7 63.6 65.2 59.4

weight is less than 4.5 kilograms. (Note: The critical values can be found in Table 1 .)

4. (25 points)

2 2

Let xl,x2, ..., x,, be an observed randoin sample from ~ ( ~ , o ~ ) , and let A ⁼⁽¹⁷-1)s loo

2 2 2 2

A size a test of HO : o So8 versus Ha : 0 > 00 is to reject Ho if A

Please find t l ~ e power function for this test.

(2)

5. (25 points)

Consider the simple linear regression model ^JI⁼

J(P +

^E⁼[l _x]

[;I +

^E and assume that these data are available: yl = 4, y2 = 3, y3 = 5, ^{y 4}⁼6 ; xl = 3, x2 = 2, x3 = 3 , x4 = 5

Find

x T x , (xTx)-' , x T y

,

j,-, ,

and (each with 5 points)

Table 1 Critical Values for the Signed-Rank Test

One-sided a = 0.01 One-sided a = 0.025 One-sided n = 0.05

71. Two-Sided a = 0.02 Two-sided a = 0.05 Two-Sided cr = 0.1

5 1

6

-

_I ₀ ₂1 2 ₄

8 2 4 6

a

3 G 8

1 0 ³ 8 I 1

1.1. 7 11 14

1.2 10 14 1 T

1 . 3 13 17 2 1

14 I lj 21 26

I. 5 20 ₂₅

no

16 2 .L ₃₀ ₃_1;

1 7 28 35 .I I.

I. S 3 3 40 4 T

18 .3 8 4 6 5.1

20 4 3 52 60

21 43 .5 9 68

2 2 .i 6 6G 7.5

23 62 ^-*i d 8 3

2 4 69 81 92

2 5

--

I I 30 1 0 :I.

2 6 ^j 95 11.0

27 33 1 07 1.20

28 1112 ₁₁₇ 1.m

2 9 111 1 2 7 141.

3 0 120 137 1.52