This is not the complete question, the complete question is:
Automobile claim amounts are modeled by a uniform distribution on the interval [0, 10,000]. Actuary A reports X, the claim amount divided by 1000. Actuary B reports Y, which is X rounded to the nearest integer from 0 to 10.Calculate the absolute value of the difference between the 4th moment of X and the 4th moment of Y
A) 0
B) 33
C) 296
D) 303
E) 533
Answer: B) 33
Step-by-step explanation:
First lets say;
z: automobile claim amounts
x: the claim amount dived by 1000
y: x rounded to the nearest integer from 0 to 10
z ≅ V[0, 10,000]
x = z / 1000 ≅ V[0, 10 ] ⇒ Fx { 1/10, 0 ≤ x ≤ 10} 0, 0/10
y = {0, 0 ≤ x < 0.5
1, 0.5 ≤ x < 1.5
2, 1.5 ≤ x < 2.5
3, 2.5 ≤ x < 3.5
↓
9, 8.5 ≤ x < 9.5
10, 9.5 ≤ x < 10
SO 4th moment of x = E(x²) = ∫₀¹⁰x⁴ 1/10 dₓ
= 1/10 (x⁵ / 5)₀¹⁰
= 10⁵ / (10 * 5)
= 100000/50
= 2000
Now
4th moment of y = E(y⁴) = ∑/y y⁴ p( y=y)
= 0⁴p( y=0) + 1⁴p( y=1 ) + 2⁴p( y=2) + → + 10⁴p( y=10)
= 0 + 1⁴.p( 0.5 ≤ x < 1.5) + 2⁴.p( 1.5 ≤ x < 2.5) + 3⁴.p( 2.5 ≤ x < 3.5 ) + → + 10⁴.p( 9.5 ≤ x < 10 )
= 1/10 [ 1⁴(1.5 - 0.5) + 2⁴(2.5 - 1.5) + → + 9⁴(9.5 - 8.5) + 10⁴(10 - 9.5)]
= 1/10 [ 1⁴ + 2⁴ + → + 9⁴ + 1/2*10⁴] = 2033.3
now the absolute difference will be
AD = ║E(x⁴) - E(y⁴)║
= ║ 2000 - 2033.3║
= 33.3 ≈ 33
