A discrete random variable 𝑋 has the following probability distribution. Find the variance of 𝑋, giving 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 that we can use to calculate the variance of 𝑋 in practice is the expectation of 𝑋 squared minus the expected value of 𝑋 squared. And we need to be clear on the difference in notation here. In the second term, we find the expected value of the discrete random variable 𝑋 and then we square it, whereas in the first term we square the 𝑋-values first and then find their expectation.
We can also recall two further formulae. First, the expected value of 𝑋 is equal to the sum of each 𝑥-value in the range of the discrete random variable, those are the values in the top row of the table we’ve been given, multiplied by their corresponding probability. Those are the values in the second row of the table. The expected value of 𝑋 squared is the sum of each 𝑥-value squared multiplied by its probability, which is inherited directly from the probability distribution of 𝑥.
We’ll find it helpful to do some of our working in a table, which is essentially an extension of the table above. In the first row, we’re going to multiply each 𝑥-value by its probability. So we have two multiplied by 0.14, which is 0.28; three multiplied by 0.25, that’s 0.75; four multiplied by 0.17, which is 0.68; five multiplied by 0.28, which is 1.4; and six multiplied by 0.16, which is 0.96. The expected value of 𝑋 is then the sum of these five values, which is 4.07.
We’ll now move on to calculating the expected value of 𝑋 squared. In the next row of our table, we’ll write down the squares of each of the 𝑥-values. The 𝑥-values are the integers two, three, four, five, and six. So their squares are the values four, nine, 16, 25, and 36. In the final row of our table, we’ll multiply these values by the probabilities. So we’re multiplying the second row of our extended table by the second row of the first table. That gives the values 0.56, 2.25, 2.72, seven, and 5.76. The expected value of 𝑋 squared is the sum of these five values, which is 18.29.
We can now calculate the variance of this discrete random variable 𝑋. We take the expected value of 𝑋 squared, that is, 18.29. And from this, we subtract the expected value of 𝑋. That’s 4.07 squared. That’s 18.29 minus 16.5649, which is 1.7251. The question specifies that we should give our answer to two decimal places, so we round up to 1.73. The variance of this discrete random variable 𝑋 then is 1.73. On its own, this doesn’t give us a huge amount of information. But we could compare this value to the variance of another discrete random variable to determine which has the greater variability.