Work out the standard deviation of the random variable 𝑥 whose probability distribution is shown. Give your answer to two decimal places.
We’ve been given the probability distribution of this discrete random variable in a graphical form. To help us answer the question, let’s convert this to a table. We’ll write the values in the range of the discrete random variable in the top row and then their associated probabilities, which are the values of 𝑓 of 𝑥, in the second row. On the graph, the values in the range of this discrete random variable are the values on the 𝑥-axis, which are one, two, three, four, and five.
The probabilities for each of these values are the height of their bar, which is the value on the 𝑦-axis. So, for example, the probability that 𝑥 is equal to one is five twelfths. The probability that 𝑥 is equal to two is three twelfths. The probability that 𝑥 is equal to three is two twelfths. And finally, the probabilities for both four and five are one twelfth. We note that the sum of these probabilities is equal to one, which should be the case for the sum of all probabilities in a probability distribution.
We’ve been asked to work out the standard deviation of this discrete random variable 𝑥, which is a measure of spread of its probability distribution. We represent this using the Greek letter 𝜎 or sometimes 𝜎 subscript 𝑥 if there are multiple variables in the same problem. The standard deviation is equal to the square root of the variants, which we write either as 𝜎 squared or var of 𝑥.
The formula for calculating the variance of a discrete random variable is as follows: it’s equal to the expectation of 𝑥 squared minus the expected value of 𝑥 squared. The difference in notation is important here. In the first term, we square the 𝑥-values first and then find their expected or average value, whereas for the second term, we find the expected value of 𝑥 and then we square this value. There’s quite a lot of work to be done here. So we need to break the calculation down into stages.
We’ll begin by calculating the expected value of 𝑥. This is equal to the sum of each value in the range of the discrete random variable multiplied by its probability. We can add a row to our table to calculate these values. First, we have one multiplied by five twelfths, which is five twelfths, then two multiplied by three twelfths, which is six twelfths. We then have three multiplied by two twelfths, which is also six twelfths, four multiplied by one twelfth, which is four twelfths, and finally five multiplied by one twelfth, which is five twelfths.
Some of these values can be simplified, but we’ll keep them all with a common denominator of 12 because we need to find their sum. We have five twelfths plus six twelfths plus six twelfths plus four twelfths plus five twelfths. That’s 26 over 12, which can be simplified to 13 over six. So we found the expected value of 𝑥.
Next, we need to compute the expected value of 𝑥 squared. The formula for this is the sum of the 𝑥 squared values multiplied by their probability and the probability distribution for 𝑥 squared is inherited directly from the probability distribution function of 𝑥. If the values in the range of the discrete random variable are one, two, three, four, and five, then the values in the range of 𝑥 squared are the squares of these values. They’re one, four, nine, 16, and 25. The probabilities for each of these values are identical to the second row in the table because the probability that 𝑥 squared is equal to four, for example, is the same as the probability that 𝑥 is equal to two.
We can add one final row to our table then, in which we multiply the values of 𝑥 squared by their associated 𝑓 of 𝑥 value. That gives five twelfths, twelve twelfths, eighteen twelfths, sixteen twelfths, and twenty-five twelfths. And again, we’ll keep each of these values with the denominator of 12 so that we can sum them easily. The sum of these five values is 76 over 12, which can be simplified to 19 over three.
Next, we calculate the variance of 𝑥, which is equal to the expected value of 𝑥 squared — that’s 19 over three — minus the expected value of 𝑥, 13 over six squared. We can use a calculator to help with this, and it gives 59 over 36. The final step is we need to take the square root of this value to give the standard deviation. So 𝜎 is equal to the square root of 59 over 36, which as a decimal is 1.2801 continuing. We’re asked to give our answer to two decimal places.
So by tabulating the probability distribution shown in the graph and then working through the various stages of the variance formula and then finding its square root, we found that the standard deviation of this discrete random variable 𝑥 to two decimal places is 1.28.