Find the mean and the standard deviation of the random variable 𝑋 for the following probability distribution. When 𝑥 equals zero, the probability equals one 12th; when 𝑥 equals two, the probability is one-sixth; when 𝑥 equals three, the probability is one-third; and when 𝑥 equals four, the probability is five 12ths.
Before starting this question, it is worth recapping four formulas. Firstly, the mean or 𝐸 of 𝑋 can be calculated by summing the values of 𝑥 multiplied by the probability of 𝑥. In a similar way, 𝐸 of 𝑋 squared can be calculated by summing the values of 𝑥 squared multiplied by the probability of 𝑥. We can use these two values to calculate the variance or Var of 𝑋. This is equal to the 𝐸 of 𝑋 squared minus the 𝐸 of 𝑋 all squared. We subtract the square of the mean from 𝐸 of 𝑋 squared. Finally, the standard deviation of 𝑋 can be calculated by square rooting the variance of 𝑋.
Let’s first begin by calculating the mean of 𝑋. This can be calculated by multiplying our 𝑥-values by our probability values and then summing the answers. Zero multiplied by one 12th is equal to zero. Two multiplied by one-sixth is equal to two-sixths or one-third. Three multiplied by one-third is equal to one. And finally, four multiplied by five 12ths is equal to 20 12ths, which can be simplified to five-thirds. One-third plus five-thirds is equal to six-thirds, which is equal to two. This means that the mean or 𝐸 of 𝑋 is equal to two plus one which equals three. The mean of the random variable 𝑋 is three.
Our next step is to calculate the value of 𝐸 of 𝑋 squared. We square our 𝑥 terms of zero, two, three, and four, multiply them by the corresponding probability values, and then sum our four answers. Zero squared is equal to zero. Multiplying this by one 12th also gives us zero. Two squared is equal to four. Multiplying this by one-sixth gives us two-thirds. Three squared is equal to nine and multiplying this by one-third gives us three. Finally, four squared equals 16 and multiplying this by five 12ths give us 20 thirds.
Adding these four numbers zero, two-thirds, three, and 20 thirds gives us 31 thirds or 10.3 recurring. We have calculated that the 𝐸 of 𝑋 squared is equal to 10.3 recurring. We can now calculate the variance of 𝑋. This is equal to the 𝐸 of 𝑋 squared minus the mean squared, in this case 10.3 recurring minus three squared. The variance is therefore equal to four-thirds.
Our final step is to calculate the standard deviation. This is the square root of the variance. We need to square root four-thirds. This is equal to two root three over three. Alternatively, the standard deviation can be written as a decimal and it is 1.1547 to four decimal places.
The mean of our random variable is three and the standard deviation is 1.1547 to four decimal places.