If the mean of a variate x is m, the mean of `(ax+b)/(c )` where a,b,c are constants is

Options

A. `(am+b)/(c )`
B. `((2b+a)/(c))m`
C. `(a+b)/(mc)`
D. none of these

Answer

To find the mean of the expression \((ax + b)/c\) given that the mean of the variate \(x\) is \(m\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mean of a Variate**:
The mean of a variate \(x\) is given as \(m\). This means that:
\[ \mu_x = m \]

2. **Express the New Variate**:
We need to find the mean of the new variate \(\frac{ax + b}{c}\). We denote this mean as \(\mu\left(\frac{ax + b}{c}\right)\).

3. **Apply the Properties of Means**:
The mean of a linear transformation of a random variable can be calculated using the formula:
\[ \mu(aX + b) = a\mu(X) + b \]
where \(a\) and \(b\) are constants.

4. **Calculate the Mean of the New Variate**:
In our case, we can rewrite the expression:
\[ \mu\left(\frac{ax + b}{c}\right) = \frac{1}{c} \mu(ax + b) \]
Now applying the linear transformation property:
\[ \mu(ax + b) = a\mu(x) + b = am + b \]
Therefore, we have:
\[ \mu\left(\frac{ax + b}{c}\right) = \frac{1}{c}(am + b) \]

5. **Final Simplification**:
This simplifies to:
\[ \mu\left(\frac{ax + b}{c}\right) = \frac{am + b}{c} \]

### Conclusion:
Thus, the mean of the variate \(\frac{ax + b}{c}\) is:
\[ \frac{am + b}{c} \]