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.
