Calculate the mean and variance of the first n natural numbers.
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Step-by-Step Solution
Step 1: Define the first n natural numbers
The set of the first n natural numbers is simply 1,2,3, up to n.
Step 2: Calculate the mean (average)
The mean of a set of numbers is the sum of all the numbers divided by the count of numbers. For the first n natural numbers, this is the sum 1+2+⋯+n divided by n.
Step 3: Use the formula for the sum of first n natural numbers
The sum of the first n natural numbers is a well-known arithmetic series, and its formula is n(n+1)/2.
Step 4: Substitute the sum into the mean formula
Substitute the sum from the previous step into the mean calculation. The n in the numerator and denominator cancels out, simplifying the expression for the mean.
Step 5: Define the variance formula
The variance measures how far each number in the set is from the mean. It is the average of the squared differences from the mean.
Step 6: Expand the variance formula
An alternative and often simpler formula for variance is the mean of the squares minus the square of the mean. This form is derived by expanding the squared term in the sum.
Step 7: Use the formula for the sum of squares of first n natural numbers
The sum of the squares of the first n natural numbers is another standard formula that will be used in our variance calculation.
Step 8: Substitute values into the variance formula
Substitute the formula for the sum of squares and the previously calculated mean into the expanded variance formula.
Step 9: Simplify the first term
The n in the numerator and denominator of the first term cancels out, simplifying it to (n+1)(2n+1)/6.
Step 10: Continue simplifying the variance expression
Now we need to subtract the square of the mean from the sum of squares. To do this, we find a common denominator, which is 12.
Step 11: Find a common denominator and combine terms
Multiply the first term by 2/2 and the second term by 3/3 to get a common denominator of 12. Then factor out the common term (n+1).
Step 12: Simplify the expression inside the brackets
Expand the terms inside the brackets and combine like terms. The expression simplifies to n−1.
Step 13: Final variance formula
Multiply (n+1) by (n−1) to get n2−1. This gives the final simplified formula for the variance of the first n natural numbers.