5. Let X be an RV with EX = 0, var(X) = 2, and EX4 = 4....

5. Let X be an RV with EX = 0, var(X) = σ2, and EX4 = μ4. Let K be any positive real number. Show that P{|X| ≥ Kσ} ≤

⎧⎪⎪⎨

⎪⎪⎩

1 if K2 < 1, 1 K2 if 1 ≤ K2 < μ4

σ4 ,

μ4−σ4

μ4+σ4K4−2K2σ4 if K2 ≥ μ4

σ4 .

In other words, show that bound (7) is better than bound (3) if K2 ≥μ4/σ4 and worse if 1 ≤ K2 < μ4/σ4. Construct an example to show that the last inequalities cannot be improved.