If X is a random variable such that `sigma(X)=2.6`, then `sigma(1-4X)` is equal to

Options

A. `7.8`
B. `-10.4`
C. 13
D. `10.4`

Answer

To find the value of \(\sigma(1 - 4X)\) given that \(\sigma(X) = 2.6\), we will follow these steps:

### Step 1: Understand the transformation
We need to find the standard deviation of the transformed random variable \(1 - 4X\). The transformation can be expressed as:
\[ Y = 1 - 4X \]
where \(Y\) is the new random variable.

### Step 2: Recall the properties of variance
The variance of a transformed variable \(AX + B\) is given by:
\[ \text{Var}(AX + B) = A^2 \cdot \text{Var}(X) \]
where \(A\) is a constant multiplier, and \(B\) is a constant added to the variable.

### Step 3: Identify \(A\) and \(B\)
In our case:
- \(A = -4\) (the coefficient of \(X\))
- \(B = 1\) (the constant added)

### Step 4: Calculate the variance of \(X\)
Given that \(\sigma(X) = 2.6\), we can find the variance of \(X\):
\[ \text{Var}(X) = \sigma^2(X) = (2.6)^2 = 6.76 \]

### Step 5: Apply the variance transformation formula
Now we apply the variance transformation formula:
\[ \text{Var}(Y) = (-4)^2 \cdot \text{Var}(X) = 16 \cdot 6.76 \]

### Step 6: Calculate the variance of \(Y\)
Now we compute:
\[ \text{Var}(Y) = 16 \cdot 6.76 = 108.16 \]

### Step 7: Find the standard deviation of \(Y\)
The standard deviation of \(Y\) is the square root of the variance:
\[ \sigma(Y) = \sqrt{\text{Var}(Y)} = \sqrt{108.16} \]

### Step 8: Calculate the final result
Calculating the square root:
\[ \sigma(Y) = 10.4 \]

Thus, the value of \(\sigma(1 - 4X)\) is \(10.4\).

### Final Answer:
\[ \sigma(1 - 4X) = 10.4 \]