1. One characteristic of a car’s storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size ��� = 5 yielded the following times:

1.1 0.9 1.4 1.1 1.0

a. Find the sample mean, ���̅.

b. Find the sample variance, ���².

c. Find the sample standard deviation, ���.

[ 1.1 ; 0.0350 ; 0.1871 ]

2. The costs of 28 “300 level” textbooks (rounded to the nearest dollar) for spring 2004, selected from all major disciplines, were as follows:

a. Group these data into six classes using as class intervals (9.50, 29.50), (29.50, 49.50), and so on.

b. Construct a histogram and interpret the histogram.

c. Find the mean, ���̅ and locate it on your histogram.

d. Find the standard deviation, ���, and locate the points ���̅ ± ��� and ���̅ ± 2��� on your histogram.

[ a.Frequencies:(7,3,4,5,5,4) ; c.67.32 ; d.35.53 ]

3. Let ��� denote the concentration of ������������3 in milligrams per liter. Twenty observations of ��� are

a. Construct an ordered stem-and-leaf display, using stems of 127, 128, …, 134.

b. Find the midrange, range, interquartile range, median, sample mean and sample variance.

[ a. Frequency:(1,1,3,1,7,4,2,1), Depths:(1,2,5,6,(7),7,3,1) b. 131.3 , 7.0 , 2.575 , 131.45 , 131.47 , 3.034 ]

4. Let the random variable ��� have the p.d.f. ���(���) = 2(1 − ���), 0 ≤ ��� ≤ 1, zero elsewhere.

a. Sketch the graph of this p.d.f.

b. Determine and sketch the graph of the distribution function of ���.

c. Find

(i) ���(0 ≤ ��� ≤ 1⁄2)

(ii) ���(1⁄4 ≤ ��� ≤ 3⁄4)

(iii) ���(��� = 3⁄4) and (iv) ���(��� ≥ 3⁄4)

0, x < 0,

[ b. F(x) = {x(2 − x), 0 ≤ x < 1,

1, 1 ≤ x.

c. (i) 3/4, (ii) 1/2, (iii) 0, (iv) 1/16.]

5. For each of the following functions, (i) find the constant ��� so that ���(���) is a p.d.f. of a random variable ���, (ii) find the distribution function, ���(���) = ���(��� ≤ ���), and (iii) sketch graphs of the p.d.f. ���(���) and the distribution function ���(���).

a.   ���(���) = 4���^���, 0 ≤ ���  ≤ 1,

b.   ���(���) = ���√���, 0 ≤ ���  ≤ 4,

c.   ���(���) = ���⁄���^3⁄4 , 0 < ��� < 1.

[ a.(i) 3,  (ii) F(x) = x⁴,  0 ≤ x ≤  1;

b.(i) 3/16,  (ii) F(x) = (1⁄8) x^3⁄2,  0 ≤ x ≤  4;

c.(i) 1/4,  (ii)  F(x) = x^1/4,  0 ≤ x ≤ 1 ]

6. For each of the distributions in Question 5, find ���, ���² and ���.

[ a. 4⁄5  ,  2⁄75  ,  √6⁄15 ;

b. 12⁄5  ,  192⁄175  ,  8 √21⁄35 ;

c. 1⁄5  ,  16⁄225  ,  4⁄15 ]

7. The p.d.f. of ��� is ���(���) = ���⁄���³,

1 < ��� < ∞.

a. Calculate the value of ��� so that ���(���) is a p.d.f.

b. Find ���(���).

c. Show that ���������(���) is not finite.

[ a. ��� = 2 ; b. ���(���) = 2 ; c. ���(���²) is unbounded. ]

8. Let ���(���) = ln ���(���), where ���(���) is the moment-generating function of a random variable of the continuous type. Show that

a. ��� = ���^′(0).

b. ���² = ���^′′(0).

9. The logistic distribution is associated with the distribution function

���(���) = (1 + ���^−���)⁻¹ , − ∞ < ��� < ∞.

Find the pdf of the logistic distribution and show that its graph is symmetric about

��� = 0.

[ ���(���) = ���−���

= ���−��� ���2��� =  ������

= ���(−���) ]

(1+���−���)2

(1+���−���)2 ���2���

(������+1)2

10. The lifetime ��� (in years) of a machine has a p.d.f.

���(���) = 2��� ���^{−(���⁄���)}2 , 0 < ���  < ∞.

���2

If ���(��� > 5) = 0.01, determine ���.

[ ��� = 2.33 ]

11. The life ��� in years of a voltage regulator of a car has the p.d.f.

���(���) =

3���²

73

���−(���⁄7)3

, 0 < ��� < ∞.