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Work out the coefficient of variation of the random variable 𝑋 whose probability distribution is shown.
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Give your answer to the nearest percent.
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We know that our figure is a probability distribution graph.
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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.
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This coefficient of variation represents how far on average data points are from the mean relative to the size of the mean.
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We will begin by calculating the mean or expected value 𝐸 of 𝑋.
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We do this by multiplying each of our 𝑋-values by the corresponding 𝑓 of 𝑥 value or probability.
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We then find the sum of all these products.
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From the graph, we begin by multiplying one by one-tenth.
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Next, we multiply three by two-tenths.
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We also need to multiply five by three-tenths and seven by four-tenths.
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Calculating each of these products gives us 0.1, 0.6, 1.5, and 2.8.
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𝐸 of 𝑋 is therefore equal to five.
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As we also need to calculate the standard deviation, our next step is to calculate 𝐸 of 𝑋 squared.
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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.
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This is equal to 0.1 plus 1.8 plus 7.5 plus 19.6.
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𝐸 of 𝑋 squared is therefore equal to 29.
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Next, we recall that the variance or var of 𝑋 is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared.
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In this question, we have 29 minus five squared.
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This is equal to four.
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Clearing some space, we have the following three values.
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We know that the standard deviation 𝜎 is equal to the positive square root of the variance of 𝑋.
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This means that in this question, the standard deviation is the positive square root of four, which equals two.
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We can now substitute our values into the formula for the coefficient of variation.
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We need to multiply two-fifths or 0.4 by 100.
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This is equal to 40 percent.
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The coefficient of variation of the random variable 𝑋 shown in the graph is 40 percent.