Solution:

Given that the mean and standard deviation of x are a and b respectively.

We need to find the SD of (x-a)/b.

Let's use the formula for SD of a random variable Y.

SD(Y) = sqrt [E(Y^2) - E(Y)^2]

Step 1: Finding E((x-a)/b)

E((x-a)/b) = (E(x) - a)/b

= (a - a)/b [since E(x) = a]

= 0/b

= 0

Step 2: Finding E(((x-a)/b)^2)

E(((x-a)/b)^2) = E((x-a)^2/b^2)

= (1/b^2) E(x^2 - 2ax + a^2)

= (1/b^2) [E(x^2) - 2aE(x) + a^2]

= (1/b^2) [E(x^2) - 2a^2 + a^2] [since E(x) = a]

= (1/b^2) [E(x^2) - a^2]

Step 3: Finding SD((x-a)/b)

SD((x-a)/b) = sqrt [E(((x-a)/b)^2) - E((x-a)/b)^2]

= sqrt [ (1/b^2) [E(x^2) - a^2] - 0^2 ]

= sqrt [(1/b^2) [E(x^2) - a^2]]

= sqrt [(E(x^2) - a^2)/b^2]

= (sqrt[E(x^2) - a^2])/b

Therefore, the SD of (x-a)/b is (sqrt[E(x^2) - a^2])/b.

Answer: Option D, a/b.

Explanation: The formula for the SD of (x-a)/b is (sqrt[E(x^2) - a^2])/b. Since E(x) = a, the term a^2 can be simplified to E(x^2). Hence, the SD of (x-a)/b is equal to the SD of x divided by b, which is a/b. Therefore, the answer is option D, a/b.