# 5 probability hw

Part 1: exercises 6.1-1, 6.1-5, 6.1-7, 6.3-3 in Probability and Statistical Inference (9th or 10th

edition).

6.1-1

6.1-1. 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 n = 5 yielded the following times: 1.1 0.9 1.4 1.1 1.0 (a) Find the sample mean, x. (b) Find the sample variance, s2. (c) Find the sample standard deviation, s.

6.1-5

In the casino game roulette, if a player bets \$1 on red, the probability of winning \$1 is 18/38 and the prob-ability of losing \$1 is 20/38. Let X equal the number of successive \$1 bets that a player makes before losing \$5. One hundred observations of X were simulated on a computer, yielding the following data:

23 127 877 65 101 45 61 95 21 43 53 49 89 9 75 93 71 39 25 91 15 131 63 63 41 7 37 13 19 413 65 43 35 23 135 703 83 49 177 61 21 9 27 507

7 17 65 7 5 87

13 213 85 83 75 95 247 1815 7 13 71 67 19 615 11 15 7 131 47 25 25 5 471 11 5 13 75 19 307 33 57 65 9 57 35 19 9 33 11 51 27 9 19 63 109 515 443 11 63 9

(a) Find the sample mean and sample standard deviation of these data.(b) Construct a relative frequency histogram of the data, using about ten classes. The classes do not need to be of the same length. (c) Locate x, x ± s, x ± 2s,and x ± 3s on your histogram. (d) In your opinion, does the median or sample mean give a better measure of the center of these data

6.1-7. Ledolter and Hogg (see References) report 64 observations that are a sample of daily weekday afternoon (3 to 7 P.M.) lead concentrations (in micrograms per cubic meter, μg/m3). The following data were recorded at an air-monitoring station near the San Diego Freeway in Los Angeles during the fall of 1976: 6.7 5.4 5.2 6.0 8.7 6.0 6.4 8.3 5.3 5.9 7.6 5.0 6.9 6.8 4.9 6.3 5.0 6.0 7.2 8.0 8.1 7.2 10.9 9.2 8.6 6.2 6.1 6.5 7.8 6.2 8.5 6.4 8.1 2.1 6.1 6.5 7.9 14.1 9.5 10.6 8.4 8.3 5.9 6.0 6.4 3.9 9.9 7.6 6.8 8.6 8.5 11.2 7.0 7.1 6.0 9.0 10.1 8.0 6.8 7.3 9.7 9.3 3.2 6.4

(a) Construct a frequency distribution of the data and dis-play the results in the form of a histogram. Is this distribution symmetric? (b) Calculate the sample mean and sample standard devi-ation.(c) Locate x, x ± s on your histogram. How many obser-vations lie within one standard deviation of the mean? How many lie within two standard deviations of the mean?

6.3-3.

Let Y1 < Y2 < Y3 < Y4 < Y5 be the order statis-tics of five independent observations from an exponential distribution that has a mean of θ = 3. (a) Find the pdf of the sample median Y3. (b) Compute the probability that Y4 is less than 5. (c) Determine P(1 < Y1).

Part 2: Suppose that you are given a dataset y1; : : : ; yn that was obtained by applying a linear

transformation to another dataset x1; : : : ; xn. Specically, for each 1 i n we have yi = axi+b

where a; b are real numbers. If x and s2

x are the sample mean and sample variance of the original

data, show that the sample mean of the transformed data y and the sample variance of the

transformed data s2y

satisfy

y = ax + b; s2y

= a2s2

x:

1