Nonnegative Random Variables
Markov's inequality applies to nonnegative random variables.
From an Average to a Probability Bound
Suppose Z is a random variable that never takes a negative value. You know its expectation, written E[Z], but you want to control the chance that Z reaches or exceeds a chosen threshold x. Markov's inequality connects these two pieces of information: the expectation and the probability of reaching the threshold.
Markov's inequality gives an upper bound. It does not generally give the exact value of the probability.
Reading the Random Variable
A nonnegative random variable is a random variable whose possible values are all at least zero. Markov's inequality requires this nonnegative condition.
The word nonnegative is essential rather than optional. The inequality is applied to a variable Z whose values do not fall below zero. Along with that variable, we need its expectation E[Z] and a threshold x. The event being bounded is Z ≥ x, meaning that Z reaches or exceeds the threshold.
Expectation as an Average
The expectation E[Z] is interpreted as the average value of the random variable Z. Markov's inequality uses this average to say how large the probability of reaching a threshold can be. A known average does not by itself specify the exact probability at that threshold; it supplies information from which an upper bound can be calculated.
Markov's Inequality
For a nonnegative random variable Z and a chosen threshold x, Markov's inequality states that the probability of Z reaching or exceeding x is at most the expectation of Z divided by x. The left side names the event being bounded. The numerator on the right is the average value, and the denominator is the threshold.
The inequality is a bound, not an equality. It says that P[Z ≥ x] cannot exceed E[Z]/x; it does not say that the two quantities must be equal.
A Bound with Actual Values
Bounding P[Z ≥ 15]
Suppose Z is nonnegative, E[Z] = 6, and the threshold is x = 15. Find an upper bound for P[Z ≥ 15].
- 1Identify the inequality
Use P[Z ≥ x] ≤ E[Z]/x because Z is a nonnegative random variable.
- 2Substitute the expectation and threshold
Replace E[Z] with 6 and x with 15, giving P[Z ≥ 15] ≤ 6/15.
- 3Simplify the ratio
The ratio 6/15 equals 0.4.
- 4Interpret the result
The probability that Z reaches or exceeds 15 is at most 0.4.
P[Z ≥ 15] ≤ 0.4
Raising the Threshold
For a fixed nonnegative random variable and a fixed expectation, Markov's bound has the form E[Z]/x. If x increases, the denominator becomes larger while the expectation stays fixed, so the ratio becomes smaller. Therefore, increasing the threshold produces a smaller or equal upper bound.
The probability P[Z ≥ x] is also monotonically nonincreasing as x increases. Reaching a higher threshold does not become more likely when the threshold is raised. Markov's inequality reflects this behavior through the decreasing ratio E[Z]/x.
When applying Markov's inequality, keep the expectation fixed in the numerator and place the selected threshold in the denominator. Then check the direction: a larger threshold should make the bound smaller or equal, not larger.
Common Calculation Errors
Common mistake Treating the Markov bound as the exact probability.
What it looks likeWriting P[Z ≥ 15] = 0.4 after calculating 6/15 = 0.4.Why it failsMarkov's inequality provides an upper bound, so the probability may be less than 0.4.
Do this insteadWrite P[Z ≥ 15] ≤ 0.4 and interpret it as the probability being at most 0.4.
Common mistake Forgetting the nonnegative condition.
What it looks likeApplying the inequality without checking whether Z can take negative values.
Why it failsMarkov's inequality applies to a random variable that never takes a negative value.
Do this insteadVerify that Z is nonnegative before using P[Z ≥ x] ≤ E[Z]/x.
Common mistake Using the threshold in the numerator.
What it looks likeReplacing E[Z]/x with x/E[Z].Why it failsThe inequality has the expectation in the numerator and the threshold in the denominator.
Do this insteadUse the form P[Z ≥ x] ≤ E[Z]/x.Common mistake Assuming a higher threshold can increase the Markov bound.
What it looks likeExpecting E[Z]/x to grow when x becomes larger.
Why it failsWith a fixed expectation, increasing the denominator makes the ratio smaller.
Do this insteadRecognize that the bound is smaller or equal for a higher threshold.
Practice the Bound
Suppose Z is a nonnegative random variable with E[Z] = 6. Use Markov's inequality to write an upper bound for P[Z ≥ 15], and state whether your result is an exact probability or a bound.
Hints
- Start with P[Z ≥ x] ≤ E[Z]/x.
- Substitute x = 15 and E[Z] = 6.
- Use at most rather than equals when interpreting the result.
Explain in words what happens to E[Z]/x when the threshold x increases while E[Z] remains fixed. Connect your explanation to the probability P[Z ≥ x].
Hints
- Focus on x as the denominator.
- A larger denominator makes the ratio smaller or equal.
- The probability of reaching a higher threshold is monotonically nonincreasing.
Key Takeaways
- 1A nonnegative random variable never takes a value below zero.
- 2Expectation E[Z] represents the average value of the random variable.
- 3For a nonnegative random variable, Markov's inequality is P[Z ≥ x] ≤ E[Z]/x.
- 4The result is an upper bound, not necessarily the exact probability.
- 5With a fixed expectation, increasing the threshold makes the Markov bound smaller or equal.
At a glance
- A nonnegative random variable has only values that are at least zero.
- Expectation describes the average value of a random variable.
- Markov's inequality connects the expectation of a nonnegative random variable to the probability that it reaches or exceeds a threshold.
- The inequality P[Z ≥ x] ≤ E[Z]/x gives an upper bound rather than an exact probability.
- Increasing the threshold decreases or leaves unchanged the resulting probability bound.