Your company, Sonic Video, Inc., has conducted research that shows the probability distribution, where X is the number of video arcades in a randomly chosen city with more than 500,000 inhabitants. x 0 1 2 3 4 5 6 7 8 9 P(X = x) 0.07 0.09 0.39 0.23 0.13 0.03 0.02 0.02 0.01 0.01 (a) Compute the mean, variance and standard deviation (accurate to one decimal place). mean variance standard deviation (b) As CEO of Startrooper Video Unlimited, you wish to install a chain of video arcades in Sleepy City, U.S.A. The city council regulations require that the number of arcades be within the range shared by at least 75% of all cities. What is this range? (Find the interval containing at least 3/4 of the data as guaranteed by Chebyshev's Rule. Round your answers to one decimal place.) (_____ ,_____) What is the largest (whole) number of video arcades you should install so as to comply with this regulation? ____________ arcades
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Your company, Sonic Video, Inc., has conducted research that shows the probability distribution, where X is the number of video arcades in a randomly chosen city with more than 500,000 inhabitants.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
|
P(X = x)
|
0.07 | 0.09 | 0.39 | 0.23 | 0.13 | 0.03 | 0.02 | 0.02 | 0.01 | 0.01 |
| mean | |
| variance | |
| standard deviation |
(b) As CEO of Startrooper Video Unlimited, you wish to install a chain of video arcades in Sleepy City, U.S.A. The city council regulations require that the number of arcades be within the range shared by at least 75% of all cities. What is this range? (Find the interval containing at least 3/4 of the data as guaranteed by Chebyshev's Rule. Round your answers to one decimal place.)
What is the largest (whole) number of video arcades you should install so as to comply with this regulation?
____________ arcades
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