Video Transcript
Let 𝑋 denote a discrete random variable which can take values one, four, and six. Given that 𝑓 of 𝑥 equals 𝑥 squared plus 11 over 𝑎, find the coefficient of variation of 𝑋. If necessary, give your answer to two decimal places.
We’ve been given the probability distribution of this discrete random variable 𝑓 of 𝑥. But it is currently in terms of an unknown value 𝑎. Before we can calculate the coefficient of variation, we’ll need to determine this value. We can do this by recalling that the sum of all the probabilities in a probability distribution function must be equal to one. So if we can find expressions for 𝑓 of one, 𝑓 of four, and 𝑓 of six — those are the probabilities for each value in the range of this discrete random variable — we can then form an equation and solve it to find the value of 𝑎.
𝑓 of one, firstly, is one squared plus 11 over 𝑎, which is 12 over 𝑎. 𝑓 of four is four squared plus 11 over 𝑎, which is 27 over 𝑎. Finally, 𝑓 of six is six squared plus 11 over 𝑎, which is 47 over 𝑎. As we’ve already said, the sum of these three values must be equal to one. So we have the equation 12 over 𝑎 plus 27 over 𝑎 plus 47 over 𝑎 is equal to one. That simplifies to 86 over 𝑎 is equal to one. And multiplying both sides of this equation by 𝑎, we find that 𝑎 is equal to 86. So we’ve determined the value of this unknown.
We can then determine the probability for each value in the range of this discrete random variable explicitly. And we’ll keep these all as fractions with a common denominator of 86 for now. Now, we’ve been asked to find the coefficient of variation for this discrete random variable 𝑋. This gives the standard deviation as a percentage of the expected value. If the discrete random variable 𝑋 has nonzero mean 𝐸 of 𝑋 and standard deviation 𝜎 sub 𝑋, then its coefficient of variation is given by 𝜎 sub 𝑋 over 𝐸 of 𝑋 multiplied by 100.
We therefore need to calculate both the expectation and the standard deviation of this discrete random variable. We have its probability distribution function, but let’s write this in a table. In the top row, we write the values in the range of this discrete random variable, which are one, four, and six. And then in the second row, we write their corresponding probabilities, which we’ve just worked out to be 12 over 86, 27 over 86, and 47 over 86.
Next, we’ll calculate the expected or average value of 𝑋. This is equal to the sum of each 𝑥-value multiplied by its probability. And we can create a new row in our table to work out these values. We have one multiplied by 12 over 86, that’s 12 over 86; four multiplied by 27 over 86, which is 108 over 86; and finally six multiplied by 47 over 86, which is 282 over 86. The expected value of 𝑋 is the sum of these three values, which is 402 over 86, or in simplified form 201 over 43.
So we have the expected value of 𝑋. And next we need to calculate the standard deviation. We recall that the standard deviation is the square root of the variance. And the variance of 𝑋 is equal to the expected value of 𝑋 squared minus the expected value of 𝑋 squared. We need to be clear on the difference in notation here. In the second term, we take the expected value of 𝑋, which we’ve just calculated, and square it, whereas in the first term we square the 𝑋-terms first and then find their expected value.
The formula for the expected value of 𝑋 squared is the sum of each 𝑥 squared value multiplied by the 𝑓 of 𝑥 values. And the 𝑓 of 𝑥 values or probabilities are inherited directly from the probability distribution of 𝑥. We can add another row to our table to find the 𝑥 squared values. The squares of one, four, and six are one, 16, and 36. And then we can add one final row in which we multiply the 𝑥 squared values by the 𝑓 of 𝑥 values, giving 12 over 86, 432 over 86, and 1692 over 86. The expected value of 𝑋 squared then is the sum of these three values, which is 2136 over 86, or in simplified form 1068 over 43.
As we’ve calculated both the expected value of 𝑋 and the expected value of 𝑋 squared, we’re now able to calculate the variance of 𝑋. It’s equal to 1068 over 43 minus 201 over 43 squared. We’ll use a calculator to evaluate this, and it gives 2.9870 continuing. The standard deviation is the square root of this value. So square rooting the exact decimal on our calculator, we have that the standard deviation of 𝑋 is equal to 1.7282 continuing.
We’ve now found both the expected value and the standard deviation of 𝑋. So all that’s left to do is to substitute these two values into the formula for the coefficient of variation. Keeping the exact value for the standard deviation of 𝑋 that we’ve just found on our calculator display, we can divide it by the expected value of 201 over 43 and then multiply by 100. Of course, dividing by 201 over 43 is the same as multiplying by 43 over 201.
Evaluating this using exact values throughout gives 36.9735 continuing. The question says that, if necessary, we should round our answer to two decimal places. So as the third digit after the decimal point is a three, we’re going to be rounding down. The coefficient of variation of 𝑋 then to two decimal places is 36.97 percent, which tells us that the standard deviation of 𝑋 is approximately 37 percent of the expected value.
