Question Video: Determining the Variance for a Discrete Random Variable Mathematics

Video Transcript

Let 𝑋 denote a discrete random variable which can take the values three, four, and five. Given that 𝑓 of 𝑥 equals 𝑎𝑥 over 12, find the variance of 𝑋. If necessary, give your answer to two decimal places.

First, we recall that the variance of a discrete random variable is a measure of the extent to which values of that discrete random variable differ from their expected value. The formula for calculating the variance of a discrete random variable 𝑋 is the expectation of 𝑋 squared minus the expectation of 𝑋 squared. And we need to be clear on the difference in notation. In the second term, we find the expected, or average, value of 𝑋 first, and then we square it, whereas in the first term, we’re finding the expected value of 𝑋 squared. So we square the 𝑋-values first and then find their expectation.

We also recall two further formulae. The expected value of 𝑋 is found by multiplying each 𝑥-value by its 𝑓 of 𝑥 value, where 𝑓 is the probability distribution of 𝑥. And we then find the sum of these values. The expected value of 𝑋 squared is similar, but this time we multiplied the squared values of 𝑥 by the 𝑓 of 𝑥 values and then find the sum. We’ve been given that 𝑓 of 𝑥 is equal to 𝑎𝑥 over 12. So we have an unknown constant 𝑎 which we need to determine. To do so, we need to recall that the sum of all probabilities in a probability distribution is equal to one. We can therefore find expressions for 𝑓 of 𝑥 for each of the three values in the range of this discrete random variable. So we can find 𝑓 of three, 𝑓 of four, and 𝑓 of five and then form an equation.

Firstly, 𝑓 of three is 𝑎 times three, or three 𝑎, over 12. And although this can be simplified, we won’t do so for now. 𝑓 of four is four 𝑎 over 12, and 𝑓 of five is five 𝑎 over 12. If the sum of these three probabilities must be one, we have the equation three 𝑎 over 12 plus four 𝑎 over 12 plus five 𝑎 over 12 is equal to one. And this is why we didn’t simplify either of the fractions that could be simplified because now they all have a common denominator of 12. Adding the three fractions, we have 12𝑎 over 12 is equal to one. And of course, the factors of 12 in the numerator and denominator cancel out, leaving 𝑎 is equal to one. The values of 𝑓 of three, 𝑓 of four, and 𝑓 of five then are three twelfths, four twelfths, and five twelfths, respectively.

We can then write the probability distribution of 𝑥 in a table with the values in the range of the discrete random variable in the first row and their corresponding probabilities in the second row. We can then add a row to our table in which we multiply each 𝑥-value by its 𝑓 of 𝑥 value. Three multiplied by three twelfths gives nine twelfths. Four multiplied by four twelfths gives sixteen twelfths. And five multiplied by five twelfths gives twenty-five twelfths. The expected value of 𝑋 is then the sum of these three values, which is fifty twelfths or 25 over six.

To find the expected value of 𝑋 squared, we need to multiply each 𝑥 squared value by 𝑓 of 𝑥, so we can add a couple more rows to our table: firstly, a row for the 𝑥 squared values, which are three multiplied by three, which is nine, four multiplied by four, which is 16, and five multiplied by five, which is 25. The final row in our table is for multiplying each 𝑥 squared value by each 𝑓 of 𝑥 value. So we have 27 over 12, 64 over 12, and 125 over 12. And again, we’ll leave all three fractions with a common denominator of 12. The expected value of 𝑋 squared is the sum of these three values. That’s 216 over 12, which is exactly equal to 18.

Finally, to find the variance of 𝑋, we take the expected value of 𝑋 squared, and from this we subtract the expected value of 𝑋 squared. So we have 18 minus 25 over six squared. That gives 18 minus 625 over 36, which as a decimal is 0.638 recurring. The question specifies that we should give our answer to two decimal places. So rounding up as the digit in the third decimal place is an eight, we have 0.64. The variance of this discrete random variable 𝑋 to two decimal places is 0.64.