Question Video: Standard Deviation of Discrete Random Variables Mathematics

Work out the coefficient of variation of the random variable 𝑋 whose probability distribution is shown. Give your answer to the nearest percent.

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Video Transcript

Work out the coefficient of variation of the random variable 𝑋 whose probability distribution is shown. Give your answer to the nearest percent.

We know that our figure is a probability distribution graph. And we recall that the coefficient of variation, written 𝐶 sub 𝑉, is equal to the standard deviation 𝜎 divided by the expected value or mean 𝐸 of 𝑋 multiplied by 100 percent. This coefficient of variation represents how far on average data points are from the mean relative to the size of the mean. We will begin by calculating the mean or expected value 𝐸 of 𝑋. We do this by multiplying each of our 𝑋-values by the corresponding 𝑓 of 𝑥 value or probability. We then find the sum of all these products.

From the graph, we begin by multiplying one by one-tenth. Next, we multiply three by two-tenths. We also need to multiply five by three-tenths and seven by four-tenths. Calculating each of these products gives us 0.1, 0.6, 1.5, and 2.8. 𝐸 of 𝑋 is therefore equal to five. As we also need to calculate the standard deviation, our next step is to calculate 𝐸 of 𝑋 squared. This is equal to one squared multiplied by one-tenth plus three squared multiplied by two-tenths plus five squared multiplied by three-tenths plus seven squared multiplied by four-tenths. This is equal to 0.1 plus 1.8 plus 7.5 plus 19.6. 𝐸 of 𝑋 squared is therefore equal to 29.

Next, we recall that the variance or var of 𝑋 is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared. In this question, we have 29 minus five squared. This is equal to four. Clearing some space, we have the following three values. We know that the standard deviation 𝜎 is equal to the positive square root of the variance of 𝑋. This means that in this question, the standard deviation is the positive square root of four, which equals two. We can now substitute our values into the formula for the coefficient of variation. We need to multiply two-fifths or 0.4 by 100. This is equal to 40 percent. The coefficient of variation of the random variable 𝑋 shown in the graph is 40 percent.

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