Calculating the Variance of (5-2x) Given SD of x is 3

Step 1: Find the Mean of (5-2x)

The mean of (5-2x) can be calculated as follows:

E(5-2x) = E(5) - 2E(x)

= 5 - 2μ (where μ is the mean of x)

Step 2: Find the Variance of (5-2x)

To find the variance of (5-2x), we need to use the formula:

Var(5-2x) = E[(5-2x)^2] - [E(5-2x)]^2

Substituting the values from step 1, we get:

Var(5-2x) = E[(5-2x)^2] - [5-2μ]^2

Step 3: Simplify the Formula

We can simplify the formula by expanding the square:

Var(5-2x) = E[25 - 20x + 4x^2] - [25 - 10μ + 4μ^2]

= E(25) - 20E(x) + 4E(x^2) - 25 + 10μ - 4μ^2

= 4E(x^2) - 20μ + 10μ - 4μ^2

= 4E(x^2) - 4μ^2 - 20μ

Step 4: Use the SD of x to Find E(x^2)

We know that SD of x is 3. So, we can use the formula:

SD(x)^2 = E(x^2) - [E(x)]^2

Substituting the values, we get:

9 = E(x^2) - μ^2

E(x^2) = 9 + μ^2

Step 5: Plug in the Values to Find Var(5-2x)

Plugging this value in the formula from step 3, we get:

Var(5-2x) = 4(9 + μ^2) - 4μ^2 - 20μ

= 36 + 4μ^2 - 4μ^2 - 20μ

= 36 - 20μ

Therefore, the variance of (5-2x) is 36 - 20μ.