Video: Determining the Variance for a Discrete Random Variable

Let 𝑋 denote a discrete random variable which can take the values 2, 3, 5, and 8. Given that 𝑃(𝑋 = 2) = 1/24, 𝑃(𝑋 = 3) = 5/12, 𝑃(𝑋 = 5) = 3/8, and 𝑃(𝑋 = 8) = 1/6, find the variance of 𝑋. Give your answer to two decimal places.

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Video Transcript

Let 𝑋 denote a discrete random variable which can take the values two, three, five, and eight. Given that the probability that 𝑋 equals two is one twenty-fourth, the probability that 𝑋 equals three is equal to five twelfths, the probability that 𝑋 equals five is three-eighths, and the probability that 𝑋 equals eight is one-sixth, find the variance of 𝑋. Give your answer to two decimal places.

We can begin by drawing a table with two rows. The top row will contain the values that the discrete random variable 𝑥 can take, in this case, two, three, five, and eight. In the bottom row, we have the corresponding probabilities. We are asked to calculate the variance of 𝑋, and we know this is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared. We can therefore begin by calculating the expected value or mean of 𝑋. 𝐸 of 𝑋 is equal to the sum of 𝑥 sub 𝑖 multiplied by 𝑝 sub 𝑖, where 𝑖 takes values from one to 𝑛. In this question, 𝑛 is equal to four as there are four possible values of 𝑥. 𝐸 of 𝑋 is therefore equal to two multiplied by one twenty-fourth plus three multiplied by five twelfths plus five multiplied by three-eighths plus eight multiplied by one-sixth. This is equal to 109 over 24.

We will now clear some space and recall how we can calculate 𝐸 of 𝑋 squared. To calculate 𝐸 of 𝑋 squared, we simply square each of our 𝑥-values and then multiply them by the corresponding probabilities. We then sum each of these values once again. 𝐸 of 𝑋 squared is equal to the calculation shown. This gives us an answer of 575 over 24. The mean of the squares 𝐸 of 𝑋 squared is equal to 575 over 24. We can now substitute in both of these values to calculate the variance. The variance of 𝑋 is equal to 575 over 24 minus 109 over 24 squared. This is equal to 1919 over 576. We are asked to give our answer to two decimal places. Therefore, the variance of the discrete random variable 𝑋 is equal to 3.33.

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