The function in the given table is a probability function of a discrete random variable 𝑋. Find the standard deviation of 𝑋. Give your answer to two decimal places.
The table tells us that the probability that 𝑥 equals four is eight 19ths. The probability that 𝑥 equals six is also eight 19ths. And the probability that 𝑥 equals 10 is three 19ths.
In order to answer this question, we need to follow four steps. Firstly, we need to calculate 𝐸 of 𝑥. Secondly, 𝐸 of 𝑥 squared. Thirdly, we will use these two answers to calculate the variance of 𝑥. And finally we’ll calculate the standard deviation of 𝑥.
𝐸 of 𝑥 is the sum of all the 𝑥-values multiplied by the 𝐹 of 𝑥 values, in this case four multiplied by eight 19ths plus six multiplied by eight 19ths plus 10 multiplied by three 19ths. This gives us a 110 19ths, or 110 divided by 19. 𝐸 of 𝑥 equals 110 divided by 19.
Our second step is to calculate 𝐸 of 𝑥 squared. 𝐸 of 𝑥 squared is equal to the sum of the 𝑥 squared values multiplied by 𝐹 of 𝑥, in this case four squared or 16 multiplied by eight 19ths, six squared multiplied by eight 19ths, and 10 squared multiplied by three 19ths.
The sum of these is 716 19ths. Therefore, 𝐸 of 𝑥 squared is equal to 716 divided by 19. Our third step is to work out the variance of 𝑥. This is calculated by subtracting the 𝐸 of 𝑥 all squared from the 𝐸 of 𝑥 squared, in this case 716 19ths minus a 110 19ths squared.
Therefore, the variance is 1504 divided by 361, or 4.166. Our final step is to calculate the standard deviation. This is calculated by square-rooting the variance. The square root of 1504 divided by 361 is equal to 2.04. Therefore, the standard deviation of the function in the table is 2.04.