Distributions of Data
Part 1.1: Probability of Burnout
Let's start with the light bulb question. We simulated the experiment of starting 100 bulbs at the same time. The data represents the number that had burned out after the first day, the second day, and so on through day 200. We reproduce some of this data here, mostly at one-week intervals through 16 weeks, in the table below. By day 200, 99 of the 100 bulbs had burned out.
Light Bulb Failure Data
|
Day
|
Bulbs burned out
|
|
Day
|
Bulbs burned out
|
|
1
|
3
|
|
49
|
79
|
|
2
|
6
|
|
56
|
83
|
|
3
|
9
|
|
63
|
86
|
|
5
|
15
|
|
70
|
89
|
|
7
|
20
|
|
77
|
91
|
|
14
|
36
|
|
84
|
93
|
|
21
|
49
|
|
91
|
95
|
|
28
|
60
|
|
98
|
96
|
|
35
|
67
|
|
105
|
97
|
|
42
|
75
|
|
112
|
97
|
- Plot the data in your worksheet.
We would like to approximate L, the unknown function whose value at time t (in days) is the number of burned out bulbs at that time, by a function LA with a simple symbolic representation. In particular, we want the graph of LA to pass through (or near) the data points given in the table.
- What about the data, or more generally about the nature of the problem, might lead you to suspect that our description of LA should contain a negative exponential?
- Plot 1 - e - rt versus t with r = 1. Now vary the magnitude of r. How does this affect the shape of the curve?
- Try to fit a function LA to the data where LA has the form
LA(t) = 100(1 - e - rt).
Explain why we need the factor of 100. Then find a value of r that makes for the best fit.
With an appropriate selection of r, we conclude that L is closely approximated by
LA(t) = 100(1 - e - rt).
-
(a) Using your factor of r from step 4, calculate LA(44). What does this number approximate?
(b) Calculate LA(5). What does this number approximate?
-
Thus, we can obtain an approximate "fractions-burned-out function" by scaling LA. If we define F(t) as LA(t)/100, then we have
F(t) =1 - e - rt.
If a and b are any numbers of days, then F(b) - F(a) approximates the fraction of bulbs burned out between time t = a and time t = b.
- Use F to find the approximate fraction of bulbs that burned out between day 10 and day 85.
Another way of looking at this is to consider what might happen if we took another bulb of the same type out of its package, screwed it into a socket, turned it on, and left it on. Our experience with the first 100 bulbs suggests it is reasonable to assert that the probability the bulb would burn out between time t = a and time t = b is F(b) - F(a).
- Find the probability that a typical bulb will burn out
