Introduction to Probability · Problem 8.2.8

Central Limit Theorem — Grinstead & Snell

Mathematics — Probability · law-of-large-numbers-continuous

Problem

Let XX be a continuous random variable and define the standardized version X∗X^* of XX by: X∗=X−μσ.X^* = \frac{X - \mu}{\sigma} . (a) Show that P(∣X∗∣≥a)≤1/a2P(|X^*| \geq a) \leq 1/a^2. (b) If XX is the random variable of Exercise 1, find bounds for P(∣X∗∣≥2)P(|X^*| \geq 2), P(∣X∗∣≥5)P(|X^*| \geq 5), and P(∣X∗∣≥9)P(|X^*| \geq 9).
Intuitive explanation

Standardization transforms any random variable into one centered at zero with unit variance—a universal scale that lets us apply powerful inequalities. Part (a) asks you to prove Chebyshev's inequality, which bounds tail probabilities using only the variance, without knowing the distribution's shape. The key is to start with the definition of variance and use the fact that ∣X∗∣≥a|X^*| \geq a is equivalent to (X∗)2≥a2(X^*)^2 \geq a^2, then apply the Markov inequality. Part (b) then uses this bound to estimate how unlikely extreme deviations are, showing how a single inequality gives useful bounds across different thresholds.

Intuitive explanation

Standardization transforms any random variable into one centered at zero with unit variance—a universal scale that lets us apply powerful inequalities. Part (a) asks you to prove Chebyshev's inequality, which bounds tail probabilities using only the variance, without knowing the distribution's shape. The key is to start with the definition of variance and use the fact that ∣X∗∣≥a|X^*| \geq a is equivalent to (X∗)2≥a2(X^*)^2 \geq a^2, then apply the Markov inequality. Part (b) then uses this bound to estimate how unlikely extreme deviations are, showing how a single inequality gives useful bounds across different thresholds.

Related theory

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