If c is a constant, then
E[X - c] = E[X] - E[c]
… = E[X] - c
and
E[(X - c)²] = E[X² - 2cX + c²]
… = E[X²] - 2c E[X] + E[c²]
… = E[X²] - 2c E[X] + c²
Recall the definition of variance:
Var[X] = E[(X - E[X])²]
… = E[X² - 2X E[X] + E[X]²]
… = E[X²] - E[2X E[X]] + E[E[X]²]
… = E[X²] - 2E[X] E[X] + E[X]²
… = E[X²] - E[X]²
The standard deviation of X is 3.2, so Var[X] = 3.2² = 10.24.
By this definition,
Var[X - c] = E[(X - c)²] - E[X - c]²
but again because c is constant,
Var[X - c] = Var[X]
This is because
Var[X - c] = E[(X - c)²] - E[X - c]²
… = E[X² - 2cX + c²] - (E[X] - E[c])²
… = E[X²] - 2c E[X] + E[c²] - E[X]² + 2 E[X] E[c] - E[c]²
… = (E[X²] - E[X]²) - 2c E[X] + E[c]² + 2c E[X] - E[c]²
… = Var[X]
Then
10.24 = 3099.2 - E[X - c]²
⇒ E[X - c]² = 3088.96
⇒ E[X - c] = 55.5784
