(i) Complete the square:
P[X ² > 2 X] = P[X ² - 2 X > 0]
P[X ² > 2 X] = P[X ² - 2 X + 1 > 1]
P[X ² > 2 X] = P[(X - 1)² > 1]
P[X ² > 2 X] = P[|X - 1| > 1]
P[X ² > 2 X] = P[X - 1 > 1] + P[-(X - 1) > 1]
P[X ² > 2 X] = P[X > 2] + P[X < 0]
P[X ² > 2 X] ≈ 0.7583
(ii) By definition of expectation, we have
E[2 X ² - 3 X - 5] = E[2 X ²] + E[-3 X] + E[-5]
E[2 X ² - 3 X - 5] = 2 E[X ²] - 3 E[X] - 5
Recall the definition of variance,
V[X] = E[(X - E[X])²] = E[X ²] - E[X]²
and that variance is the square of the standard deviation. So we have
E[2 X ² - 3 X - 5] = 2 (V[X] + E[X]²) - 3 E[X] - 5
E[2 X ² - 3 X - 5] = 2 (2² + 3²) - 3² - 5
E[2 X ² - 3 X - 5] = 12
