If s.d. of X is `sigma` then s.d. of the variable `U=(aX+b)/(c )` where a,b,c are constants is

Options

A. `|c/a|sigma`
B. `|a/c|sigma`
C. `|b/c|sigma`
D. `(c^(2))/(a^(2))sigma`

Answer

To find the standard deviation of the variable \( U = \frac{aX + b}{c} \) where \( a, b, c \) are constants and the standard deviation of \( X \) is \( \sigma \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Transformation**:
The variable \( U \) is a linear transformation of the variable \( X \). The general form of a linear transformation is given by:
\[ U = \frac{aX + b}{c} \]

2. **Standard Deviation of a Linear Transformation**:
The standard deviation of a linear transformation of a random variable can be expressed as:
\[ \text{s.d.}(U) = \left| \frac{a}{c} \right| \cdot \text{s.d.}(X) \]
This means that the standard deviation of \( U \) is scaled by the factor \( \frac{a}{c} \).

3. **Substituting the Known Values**:
Since the standard deviation of \( X \) is given as \( \sigma \), we can substitute this into the equation:
\[ \text{s.d.}(U) = \left| \frac{a}{c} \right| \cdot \sigma \]

4. **Final Result**:
Therefore, the standard deviation of the variable \( U \) is:
\[ \text{s.d.}(U) = \left| \frac{a}{c} \right| \sigma \]

### Summary:
The standard deviation of the variable \( U = \frac{aX + b}{c} \) is given by:
\[ \text{s.d.}(U) = \left| \frac{a}{c} \right| \sigma \]