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.
