The table lists the number of wireless devices per household i...

Question

The table lists the number of wireless devices per household in a small town in the United States.

Table showing the number of wireless devices per household and corresponding household counts in a small town.

c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.

Accepted Answer

All right, hello, everyone. So this question is asking us to find the mean variance and standard deviation for the following probability distribution. So to begin, let's go ahead and find our mean first. So, mu, if you recall, that's the mean, is equal to the sum of all X values multiplied by their respective probability or their respective P X value. So, for example, you would add together 0 multiplied by its probability of 0.19, oops. Added to 1 multiplied by 0.34. Added to 2 multiplied by 0.26. Added to 3 multiplied by 0.13. Added to 4 multiplied by 0.06. And lastly, added to 5 multiplied by 0.02. And so evaluating this expression gives you your mean mu equal to 1.59. So now, it's this value that you're going to need to calculate the variance and then eventually the standard deviation. You'll see here that I went ahead and made a table with all X values given in their respective probabilities. Now the very right column of this table has been left blank. Or the calculations necessary to carry out the Or rather to find the variants, but there's actually one column that I haven't put here yet, so let me go ahead and do that. Now the reason why It was necessary to find the mean first. Is because now for each value of X. You would subtract the mean from each value of X and then subsequently square it. And then The value that you obtain from. Squaring the difference between X and the mean, you would then multiply by its respective probability. So let's begin, for example, with 0. What you do first is you take the X value of zero and you subtract that by the mean, which is 1.59. This gives you negative 1.59. Now, -1.59 squared equals 2. 5281 So now, 2.5281 is what you would multiply by the probability. Of X equals 0, which in this case is 0.19. So 2.5281, multiplied by 0.19, equals 0.4803. So it's this same sequence of steps that you would repeat for the remaining X values. And so once you repeat the necessary calculations for the remaining X values, the table that I made here is now full. And it's at this point that you have the information necessary to calculate the variance. Now, the variance or sigma squared is equal to the sum of. The differences between each X-value and the mean squared multiplied by each Xval's respective probability. So With the calculations you've already made. The variance Is equal to the sum of. All of the values on this very right column of the table that I have on the screen. So that's 0.4803. Added to 0.1184. 0.0437, 0.2585, 0.3485, and 0.2326. The sum of all of these values together comes out to 1.4820. Which corresponds to your variants. So from the variants, you can now find the standard deviation or sigma. Note that the standard deviation is simply the square root of the variance. So sigma is equal to the square root of 1.4820. And that gives you approximately 1.22 after rounding. And there you have it. So now that completes our final answer. The mean is equal to 1.59, the variance 1.48, and the standard deviation 1.22. So with that being said, thank you so very much for watching, and I hope you found this helpful.

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The table lists the number of wireless devices per household in a small town in the United States.

c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.

Key Concepts

Mean

Variance

Standard Deviation

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The table lists the number of wireless devices per household in a small town in the United States.c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.