if a variable takes values 0,1,2..........,n with frequencies 1,nc1,........ncn then the AM is

To find the arithmetic mean (AM) of a variable that takes values from 0 to n with specific frequencies, we can follow a systematic approach. The values and their corresponding frequencies are crucial in calculating the mean accurately. Let's break this down step by step.

Understanding the Variables and Frequencies

In this scenario, we have a variable that can take on values from 0 to n. The frequencies associated with these values are as follows:

• Value 0 has a frequency of 1
• Value 1 has a frequency of nC1 (which is the number of combinations of n items taken 1 at a time)
• Value 2 has a frequency of nC2
• ...
• Value n has a frequency of nCn (which is 1, as it represents choosing all n items)

Calculating the Total Frequency

The total frequency (N) can be calculated by summing all the individual frequencies:

N = 1 + nC1 + nC2 + ... + nCn

This sum is equal to 2^n, based on the binomial theorem, which states that the sum of the coefficients in the expansion of (1 + 1)^n is 2^n.

Finding the Weighted Sum of Values

Next, we need to compute the weighted sum of the values, which is done by multiplying each value by its corresponding frequency:

Weighted Sum = (0 * 1) + (1 * nC1) + (2 * nC2) + ... + (n * nCn)

This can be simplified as:

Weighted Sum = 0 + nC1 + 2nC2 + ... + nnCn

Using the Binomial Theorem

To evaluate the weighted sum, we can utilize the binomial theorem again. The expression for the weighted sum can be derived from the expansion of (1 + x)^n, where we differentiate it:

When we differentiate (1 + x)^n and then multiply by x, we get:

x * d/dx[(1 + x)^n] = nx(1 + x)^(n-1)

Evaluating this at x = 1 gives us:

Weighted Sum = n * 2^(n-1)

Calculating the Arithmetic Mean

Now that we have both the weighted sum and the total frequency, we can find the arithmetic mean:

AM = (Weighted Sum) / N

Substituting the values we found:

AM = (n * 2^(n-1)) / (2^n)

By simplifying this expression, we get:

AM = n / 2

Final Result

Thus, the arithmetic mean of the variable that takes values from 0 to n with the specified frequencies is:

AM = n / 2

This result shows that the mean is simply half of the maximum value n, which is a common outcome in uniformly distributed variables over a defined range. If you have any further questions or need clarification on any part of this process, feel free to ask!