For the following probability distribution, find the mean, the standard deviation, and the coefficient of variation of the variable 𝑋.
Before we start, let’s think about these words. The mean of this data is found by summing 𝑋 times the probability of 𝑋 for every value. From there, we look for the standard deviation. Standard deviation, which we usually label with the 𝛿, is the square root of the variance of 𝑋. And that means we’ll need to find the variance of 𝑋. To find the variance, we take the summation of 𝑋 squared times the probability of 𝑋 for every value of 𝑋. And then, we subtract the mean squared, which we know is the summation of 𝑋 times the probability of 𝑋 for every 𝑋 value. And finally, we need the coefficient of variation, which is equal to the standard deviation over the mean.
Let’s start with the mean. Remember, to find the mean, we need to sum 𝑋 times the probability of 𝑋 for every value of 𝑋. And in this probability distribution, the 𝑓 of 𝑋 of 𝑟 is our probability. So we multiply one times 0.1, which equals 0.1. And then, we add two times 0.2, 0.4 plus three times 0.3, 0.9 plus four times 0.4, which is 1.6. When we add those together, we see that the mean equals three. And while the next thing we want to find is our standard deviation, we can’t find that without first finding the variance. And to find the variance, we need the mean. But we also need the sum of 𝑋 squared times the probability of 𝑋. And the first part of that is simply finding 𝑋 squared. One squared equals one. Two squared equals four. Three squared equals nine. And four squared equals 16. Notice what’s happened here. We’ve taken the values from row one and squared them to create this 𝑋 squared row.
Now that we know that, we’re gonna need to multiply each of these 𝑋 squared terms by the probability of that 𝑋. We’re multiplying the values in the second row by the values in the fourth row. 0.1 times one equals 0.1. 0.2 times four equals 0.8. 0.3 times nine equals 2.7. And 0.4 times 16 equals 6.4. And then, we need to find the summation of these values. We need to add them together. The summation of 𝑋 squared times the probability of 𝑋 equals 10.
Looking back at the variance formula, this is the part that will equal 10. For us, the variance of 𝑋 equals 10 minus the mean squared. We know that the mean for us is three. 10 minus three squared equals 10 minus nine. The variance of 𝑋 equals one. And the standard deviation is the square root of the variance. Our standard deviation equals the square root of one, which is one. And at this point, we have all the information we need to find the coefficient of variation. The coefficient of variation equals the standard deviation over the mean. For us, that’s one over three.