Find the mean and the standard deviation of the random variable 𝑥 for the given probability distribution.
The probability distribution of a random variable tells us the values that that random variable can take and their associated probabilities. In this question, we can see that the random variable 𝑥 can take the values zero, one, two, and three with associated probabilities of one-sixth, one twelfth, one-third, and five twelfths. Notice that the probabilities for all values must sum to one. And if we convert these fractions to fractions with a common denominator of 12, we can see that this is indeed the case for the probabilities in this question.
We’ve been asked to find the mean and the standard deviation of the random variable 𝑥. And, in fact, there are standard formulae and methods that we can apply. To calculate the mean, 𝜇, of a random variable, we need to find the sum of each value multiplied by its associated probability. So we have zero multiplied by one-sixth plus one multiplied by one twelfth plus two multiplied by one-third plus three multiplied by five twelfths.
The first term is just equal to zero. And then we have one twelfth plus two-thirds plus fifteen twelfths. To add these three fractions, we need a common denominator of 12. So we convert the fraction two-thirds to a fraction over 12 by multiplying both the numerator and denominator by four. We now have one twelfth plus eight twelfths plus fifteen twelfths. This gives twenty-four twelfths. And 24 over 12, or 24 divided by 12, is exactly equal to two. So we found the mean, 𝜇, of this random variable 𝑥; it’s two.
Before we can calculate the standard deviation, 𝛿, of the random variable 𝑥, we must first find its variance, 𝛿 squared. And to do so, we square each of the values 𝑥 can take and multiply them by their associated probability. Then, find the sum of these values. We then subtract the mean squared. So we have zero squared multiplied by one-sixth plus one squared multiplied by one twelfth plus two squared multiplied by one-third plus three squared minus five twelfths. And then we subtract 𝜇 squared. And we’ve just found that 𝜇 is equal to two. So we’re subtracting two squared.
Now, for the first term, this is just equal to zero. For the second, one squared is one, and multiplying by one twelfth just gives one twelfth. For the third term, two squared is four, and multiplying by one-third gives four-thirds. For the fourth term, three squared is nine, and multiplying by five twelfths give forty-five twelfths. Then two squared is four. So we have one twelfth plus four-thirds plus forty-five twelfths minus four.
Again, we need a common denominator of 12 in order to add these fractions. So we have one twelfth plus sixteen twelfths, that’s multiplying the numerator and denominator by four, plus forty-five twelfths minus forty-eight twelfths. That’s multiplying the numerator of four and the hidden denominator of one by 12. This gives the fraction fourteen twelfths. And dividing numerator and denominator by two, this simplifies to seven-sixths.
But, remember, this gives the variance of the random variable 𝑥, 𝛿 squared. So to find the value of 𝛿, we need to take the square root of seven-sixths. We can evaluate this on a calculator, and it gives the decimal 1.0801. It’s usual to round decimal values like this to three significant figures. So we have that the mean, 𝜇, of the distribution is two. And the standard deviation, 𝛿, is 1.08, to three significant figures.