In Exercises 7 and 8, (a) find the mean, variance, and standar...

Question

In Exercises 7 and 8, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.

The number of cell phones per household in a small town

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Accepted Answer

All right. Hello, everyone. This question says, a factory produces light bulbs, and the number of defective bulbs found in a random sample of 5 bulbs is given by the following probability distribution. Find the mean variance and standard deviation of the probability distribution. So, just for the sake of clarity, let's say that X is the number of defective bulbs, and P X is the corresponding probability listed in the table. Now the formula for the mean or mu. is equal to the sum of each value of X multiplied by its respective probability. So let's begin with calculating the mean first, because we already have each X value and each corresponding P of X. So, the mean is equal to the sum of 0 multiplied by 0.123, which is its probability. 1 multiplied by 0.317. 2 multiplied by 0.282. 3 multiplied by 0.178. 4 multiplied by 0.074. And 5 multiplied by 0.026. So, evaluating this expression, the mean is ultimately equal to approximately 1.8. And it's this value of the mean that's going to be relevant for the table that I have up on the screen, but let me actually move it upwards a little bit. OK. So here On the screen, I have a table where I've already written each X value and each corresponding P of X. Now, the reason why it was necessary to calculate the mean first is for the contents of these two blank columns that I have here towards the right. So the 2nd to last column of the table. Is going to be Where you find the difference between each X value and the mean, and then square that value. Then for the next column over, you take what you got. In the second to last column and multiply it by its respective probability for that particular X value. So let's use 0 here as an example. Right, what you do first is take X and subtracted by mu, which is the mean. So, 0, subtracted by 0 by 1.8, excuse me, is -1.8. Squaring this value gives you. 3.3893. Which is what I would write down in the 2nd to last column of my table. So now, 3.3893 is what I multiply. By the probability of X equals 0, which is 0.123. So, 3.3893, multiplied by 0.0123 or Sorry, 0.123. Gives me 0.416. 882 Which is the number I write down on the last column of my table. And from here you would repeat this calculation for X1 through 5. So This is what the table should look like after repeating the calculations for X equals 1 through X equals 5. So now, it's the information in the very last column that's going to be necessary for finding the variants, because recall that the variance or sigma squared is equal to the sum of All X values or each X value, subtracted by mu squared multiplied by P X. So it's going to be the sum of all the values obtained in this last column of the table. When you add all of these components together. Sigma squared, that's the variance. Is ultimately equal to approximately 1.5 after rounding. So now Now that you have the variance, recall that sigma, that's the standard deviation, is simply the square root of the variance. So the standard deviation is equal to the square root of 1.5 Which is approximately 1.2. So now, we've arrived at our final answer. As mentioned before, mu is equal to approximately 1.8. Sigma squared or the variance is 1.5, and sigma or the standard deviation is 1.2. And so with that being said, thank you so very much for watching, and I hope we found this helpful.

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In Exercises 7 and 8, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.

The number of cell phones per household in a small town

Key Concepts

Mean of a Probability Distribution

Variance and Standard Deviation

Interpreting Results

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In Exercises 7 and 8, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.The number of cell phones per household in a small town