What is Markov's inequality?

Markov's inequality states that for any non-negative random variable X and positive a: P(X ≥ a) ≤ E[X] / a. It provides an upper bound on tail probability using only the expected value, without knowing the full distribution.

When is Markov's inequality useful?

Markov's inequality is useful when you only know E[X] and need a guaranteed upper bound on P(X ≥ a). It's most informative when a is significantly larger than E[X], making the bound less than 1.

Why does the tool show a warning when a ≤ E[X]?

When a ≤ E[X], the Markov bound E[X]/a becomes ≥ 1. Since probabilities can't exceed 1, a bound greater than 100% tells us nothing. The inequality only provides useful information when a > E[X].

Why is the Markov bound often much larger than actual probability?

Markov's inequality must hold for ANY non-negative distribution with that expected value. The worst-case distribution determines the bound. Well-behaved distributions like Normal or Exponential have actual tail probabilities much smaller than the bound.

Markov Inequality Visualization

Markov Inequality Visualizer

P(X ≥ a) ≤ E[X] / a

P(X ≥ 15) ≤ 10 / 15 = 0.667 = 66.7%

Actual: 22.3%

Distribution Type

E[X]: 10

Threshold a: 15

Markov Inequality: For non-negative X and a > 0: P(X ≥ a) ≤ E[X] / a

Red area shows P(X ≥ a). Bound: 66.7%, Actual: 22.3%

Getting Started with the Markov Visualizer

Understanding the Markov Bound

Using the Distribution Selector

Adjusting E[X] and Threshold

When Markov Becomes Useless

Comparing Bound to Actual Probability

Why Markov's Inequality Matters

Related Tools and Concepts

Visualizing Markov's Inequality

Markov's inequality states that for any non-negative random variable X and positive threshold a: P(X ≥ a) ≤ E[X] / a. This tool visualizes the bound across nine distributions, showing both the Markov bound and the actual tail probability. Adjust E[X] and threshold to see when bounds are tight versus loose.

Getting Started with the Markov Visualizer

This tool demonstrates Markov's inequality, which bounds tail probabilities using only the expected value. The visualization shows a probability distribution with the tail region P(X ≥ a) highlighted in red.

The left panel displays the PDF (for continuous) or PMF (for discrete distributions). A green dashed line marks E[X], and a red dashed line marks the threshold a. The red shaded area (or red bars) represents the actual tail probability.

The top panel shows the Markov bound formula and compares it to the actual probability. Adjust E[X] and threshold to explore when the bound is tight versus loose across different distributions.

Understanding the Markov Bound

Markov's inequality states:

P(X \geq a) \leq \frac{E[X]}{a}

This bound requires only two conditions:
• X must be non-negative (X ≥ 0 always)
• a must be positive (a > 0)

The bound decreases as a increases relative to E[X]. When a = 2·E[X], the bound is 0.5 (50%). When a = 10·E[X], the bound is 0.1 (10%).

The visualization shows both the Markov bound (theoretical maximum) and the actual tail probability. For most distributions, the actual probability is much smaller than the bound, demonstrating that Markov's inequality is often conservative.

Using the Distribution Selector

Nine distributions are available, grouped into continuous and discrete:

Continuous Distributions:
• Normal (positive range displayed)
• Exponential
• Uniform

Discrete Distributions:
• Poisson
• Binomial
• Geometric
• Negative Binomial
• Hypergeometric
• Discrete Uniform

Each distribution shows different tail behavior. Exponential has a long right tail, making Markov relatively tight. Normal concentrates around the mean, making the bound very loose. Discrete distributions display as vertical bars with dots at PMF values.

Try switching distributions while keeping E[X] and threshold fixed to see how the same Markov bound applies differently to different shapes.

Adjusting E[X] and Threshold

Two sliders control the key parameters:

E[X] Slider (green): Sets the expected value from 1 to 30. This changes the distribution's location and scale. Higher E[X] generally spreads the distribution rightward.

Threshold (a) Slider (red): Sets the threshold from 1 to 40. This determines where tail probability is measured. The red dashed line and shaded region update accordingly.

Key experiments to try:

• Set a = E[X]: The bound becomes 1 (100%), which tells us nothing
• Set a = 2·E[X]: The bound becomes 0.5 (50%)
• Set a >> E[X]: The bound becomes small, and actual probability becomes tiny
• Compare Exponential vs Normal at the same settings

When Markov Becomes Useless

The visualization displays a warning when a ≤ E[X]. In this case:

\frac{E[X]}{a} \geq 1

A probability bound of 100% or more is trivially true and provides no information. Every probability is ≤ 1, so saying P(X ≥ a) ≤ 1.5 tells us nothing.

The warning box turns red and explains that the bound is useless. To get meaningful information, increase a above E[X].

This limitation is fundamental to Markov's inequality. The bound only constrains tail probabilities in the region beyond the expected value. For probabilities closer to the center, you need stronger inequalities like Chebyshev.

Comparing Bound to Actual Probability

The information panel shows two values:

• Bound: The Markov upper bound E[X]/a
• Actual: The true tail probability P(X ≥ a)

The ratio Bound/Actual indicates how loose the inequality is. Typical observations:

Exponential distribution: Bound is relatively tight (ratio 2-5x)
Normal distribution: Bound is very loose (ratio 10-100x or more)
Uniform distribution: Bound can be exact at certain thresholds

The gap exists because Markov must hold for ANY non-negative distribution with that E[X]. The worst-case distribution (which achieves the bound) places all probability mass at exactly 0 and a, creating maximum tail probability.

Real distributions spread probability more evenly, giving smaller tails than the worst case.

Why Markov's Inequality Matters

Despite being loose, Markov's inequality is valuable because:

Minimal requirements: Only needs X ≥ 0 and E[X] known. No variance, no distribution shape.

Universal applicability: Works for any non-negative random variable, continuous or discrete.

Theoretical foundation: Building block for stronger inequalities. Chebyshev's inequality is derived from Markov applied to (X - μ)².

Quick bounds: When you only know the average, Markov gives an instant upper bound.

Applications include algorithm analysis (bounding worst-case by average-case), queueing theory (service time bounds), and proving convergence in probability (weak law of large numbers).

Related Tools and Concepts

Markov's inequality connects to other probability concepts and tools:

Theory Pages:

• Markov Inequality covers complete theory and proofs

• Chebyshev Inequality provides tighter bounds using variance

• Expected Value is the key input for Markov

• Variance enables stronger bounds

Other Visualizations:

• Chebyshev Visualizer shows two-tailed deviation bounds

• Expected Value Visualizer demonstrates E[X] concepts

• Distribution Visualizers display full PDFs and CDFs

Related Topics:

• Probability Distributions covers the distributions used here

• Central Limit Theorem uses concentration concepts