Final Answer:
The expected value of E((X-2)²) is 8. This represents the average of the squared deviations of the random variable X from the value 2.
Step-by-step explanation:
The question involves finding the expected value of the square of the random variable X minus 2, i.e., E[(X - 2)²]. The expected value, denoted by E, represents the average value a random variable would take over many repetitions of an experiment. For a continuous random variable X, E(X) is computed as the integral of X multiplied by its probability density function over its entire range.
In this specific case:
- X is a random variable.
- E(X) is the expected value of X and is given as 4.
- E(X²) is the expected value of X^2 and is given as 20.
To find E((X-2)²), we need to expand and simplify the expression.
(a) Expand the expression:
(X-2)² = X² - 4X + 4
(b) Use the linearity of expectation:
E((X-2)²) = E(X² - 4X + 4)
(c) Apply the properties of expectation:
E(X² - 4X + 4) = E(X²) - E(4X) + E(4)
(d) Given that E(X²) = 20 and E(X) = 4:
E(X²) - E(4X) + E(4) = 20 - 4E(X) + 4
(e) Substitute the value of E(X) = 4:
20 - 4E(X) + 4 = 20 - 4(4) + 4
= 20 - 16 + 4
= 8
Therefore, E((X-2)²) = 8.