265 IOE Fall 07 September 20, 2007
Lab 2: Random Number Distributions
Note: Please make sure you type in your answers in the spaces provided and submit this file electronically via the CTools website under Assignments. Save this file as `uniqname_lab_2' with your own uniqname.
Section 1: Examples of Discrete Random Variables
1. On your own, name three examples of a discrete random variable: (1 point) 1. Measurements of a digital instrument that displays the current to the nearest one-hundredth of a milliampere 2. A voice communication system containing 48 external lines 3. Number of nonzero digits in a randomly selected zip code. 2. Pretend you are rolling a pair of fair dice (meaning each face turning up is equally likely); however, these are not standard dice. One of the dice has 10 faces (number 1 through 10), and the other has only 4 faces (numbered 1 through 4). What is the probability of the two dice totaling 8 on a given roll? (1 point) 10% Now, in the box below, paste a histogram (from either Excel or Minitab) that displays the probability of rolling any value between 2 and 14. Comment briefly on the shape of this distribution: (1 point)
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Histogram of total
20
15
Percent
10
5
0
2
3
4
5
6
7 8 total
9
10
11
12
13
14
The Histogram is not symmetrical. There are no outliers on the graph either. The midpoint is between 7 and 8. The range of this graph is from 2-14. 3. Since moving into East Quad, Teddy has been receiving a lot of mail from his friends and family. On any given day, there is a 50% probability he will receive a letter from his mom, an 80% chance he will receive a letter from his girlfriend, and a 90% chance he will receive a letter from his pen pal. What is the probability that Teddy will receive 2 letters on a given day? (1 point) 49% Now, in the box below, paste a PMF (histogram from either Excel or Minitab) of how many letters Teddy will receive on a given day. Comment briefly on the shape of this distribution: (1 point)
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Histogram of Letters
50
40
Percent
30
20
10
0
0
1 Letters
2
3
The histogram is not symmetric. It is skewed towards the right because he has very little chance of receiving zero letters. range The is from 0 to 3.
Section 2: Dirtbike Analysis
To answer the following questions, please see data file `Dirtbike_Data.csv'. 4. What is the mean number of gears (see column `transmission') across all dirtbikes? What is the standard deviation? (0.5 point) a) 4.957 b) 1.325 What is the mean compression ratio (see column `comp.ratio') across all dirtbikes? What is the standard deviation? (0.5 point) c) 9.917 d) 1.599
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5. Assume that we are drawing one dirtbike at random from this population of 93 dirtbikes. Create a PMF for the number of gears we would find on this randomly chosen bike and paste it below. Now, create a CDF of the same gear data (see Graph... Empirical CDF... in Minitab), and paste it below as well. Comment on the connection in the shape of these two graphs: (2 points)
Histogram of transmission
50
40
Percent
30
20
10
0
1
2
3 4 transmission
5
6
paste CDF
4
Empirical CDF of transmission
100
80
Percent
60
40
20
0 1 2 3 4 transmission 5 6
The histogram is skewed more toward the right showing that most of the dirt bikes use higher gear settings of 5 and 6. The CDF graph corresponds to the area under the bars of the histogram. So if you were to add up the percent of the bars one through three in the histogram you would get approximately 20% as shown in the CDF. The higher the slope on the CDF means a higher peak on the graph of the PMF and vice versa. 6. Repeat Question 5, except this time create a PMF and CDF of the compression ratio data and paste it below. Again, comment on the connection in the shape of these two graphs: (2 points)
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Histogram of comp.ratio
35 30 25
Percent
20 15 10 5 0
6
7
8
9 10 comp.ratio
11
12
13
Empirical CDF of comp.ratio
100
80
Percent
60
40
20
0 6 7 8 9 10 comp.ratio 11 12 13
The histogram is not symmetric and is neither skewed to the right or left. The CDF graph again shows the area under the bars of the histogram and if you added up all the
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percentages up to 10 on the histogram, you would get 40% like shown in the CDF. Finally the higher the slope of the CDF the greater the peak in the PMF, which again can be seen by looking at the value of 10 on both graphs.
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