If the variable takes values `0,1,2,…, n` with frequencies `q^(n),""^(n)C_(1)q^(n-1)p, ""^(n)C_(2)q^(n-2)p^(2), ... ""^(n)C_(n)p^(n), " where " p+q=1`, then the mean is

Options

A. A. np
B. B. nq
C. c .`n(p+q)`
D. d. none of these

Answer

To find the mean of the given variable with specified frequencies, we can follow these steps:

### Step 1: Understand the frequencies
The frequencies of the variable taking values \(0, 1, 2, \ldots, n\) are given by:
- For \(r = 0\): \( \binom{n}{0} q^n p^0 = q^n \)
- For \(r = 1\): \( \binom{n}{1} q^{n-1} p^1 = n q^{n-1} p \)
- For \(r = 2\): \( \binom{n}{2} q^{n-2} p^2 = \frac{n(n-1)}{2} q^{n-2} p^2 \)
- ...
- For \(r = n\): \( \binom{n}{n} q^0 p^n = p^n \)

### Step 2: Write the expression for the mean
The mean \( \mu \) can be calculated using the formula:
\[ \mu = \frac{\sum (x \cdot f)}{\sum f} \]
where \(x\) is the value and \(f\) is the frequency.

### Step 3: Set up the summation for the mean
We need to compute:
\[ \mu = \frac{\sum_{r=0}^{n} r \cdot \binom{n}{r} q^{n-r} p^r}{\sum_{r=0}^{n} \binom{n}{r} q^{n-r} p^r} \]

### Step 4: Calculate the denominator
The denominator is the sum of the frequencies:
\[ \sum_{r=0}^{n} \binom{n}{r} q^{n-r} p^r = (p + q)^n = 1^n = 1 \]

### Step 5: Calculate the numerator
The numerator can be simplified using the property of binomial coefficients:
\[ \sum_{r=0}^{n} r \cdot \binom{n}{r} q^{n-r} p^r = n \cdot \sum_{r=1}^{n} \binom{n-1}{r-1} q^{n-r} p^r \]
This can be rewritten as:
\[ n \cdot p \cdot (p + q)^{n-1} = n \cdot p \cdot 1^{n-1} = n \cdot p \]

### Step 6: Combine results to find the mean
Now, substituting back into the mean formula:
\[ \mu = \frac{n \cdot p}{1} = n \cdot p \]

### Final Answer
Thus, the mean of the variable is:
\[ \mu = n \cdot p \]