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For 40 values of the variable x, it is given that E(x-c)² = 3099.2, where is a constant. The standard deviation of these values of x is 3.2.

(i) Find the value of E(x-c).

Sagot :

LammettHash

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

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