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 0.36 0.24 0.15 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.)
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 | 0.36 | 0.24 | 0.15 | 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.)
(a)
Consider the probability distribution for the random variable X.
Steps to calculate the standard deviation using Excel Spread sheet as follows:
Step1. Name cells A1, and B1 as x, and P(X=x) respectively.
Step2. Enter the values of x through cells A2 to A11.
Step3. Enter the values of P(X=x) through cells B2 to B11.
Step4. Name the cell C1 as xP(X=x). Click on C2 and add the formula “=A2*B2” in the formula tab and drag the formula to the cells through C2 to C11.
Step5. Name the cell C12 as mean (mu).
Step6. Click on C13 and add the formula “=SUM(C2:C11)” in the formula tab.
Step7. Click on D2 and add the formula “=(A2-$C$13)^2*B2” in the formula tab and drag the formula to the cells through D2 to D11.
Step8. Name the cell D12 as variance.
Step11. Click on D13 and add the formula “=SUM(D2:D11)” in the formula tab.
Step8. Name the cell E1 as SD (Standard Deviation).
Step7. Click on E13 and add the formula “=SQRT(D13)” in the formula tab.
A window showing the standard deviation of the sample will appear as,
|
x |
P(X=x) |
xP(X=x) |
(x-mu)^2P(X=x) |
SD |
|
0 |
0.07 |
0 |
0.514087 |
|
|
1 |
0.09 |
0.09 |
0.263169 |
|
|
2 |
0.36 |
0.72 |
0.181476 |
|
|
3 |
0.24 |
0.72 |
0.020184 |
|
|
4 |
0.15 |
0.6 |
0.249615 |
|
|
5 |
0.03 |
0.15 |
0.157323 |
|
|
6 |
0.02 |
0.12 |
0.216482 |
|
|
7 |
0.02 |
0.14 |
0.368082 |
|
|
8 |
0.01 |
0.08 |
0.279841 |
|
|
9 |
0.01 |
0.09 |
0.395641 |
|
|
mean(mu) |
Variance |
|||
|
2.71 |
2.6459 |
1.6266 |
Thus, the mean of random variable X is 2.71, variance is 2.6 and the standard deviation is σ≈1.6.
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