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
P[Z ≥ x] ≤ E[Z] / xFor 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].
Identify the inequality: Use P[Z ≥ x] ≤ E[Z]/x because Z is a nonnegative random variable.
Substitute the expectation and threshold: Replace E[Z] with 6 and x with 15, giving P[Z ≥ 15] ≤ 6/15.
Simplify the ratio: The ratio 6/15 equals 0.4.
Interpret 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
Treating the Markov bound as the exact probability.
Markov's inequality provides an upper bound, so the probability may be less than 0.4.
Fix:
Write P[Z ≥ 15] ≤ 0.4 and interpret it as the probability being at most 0.4.Forgetting the nonnegative condition.
Markov's inequality applies to a random variable that never takes a negative value.
Fix:
Verify that Z is nonnegative before using P[Z ≥ x] ≤ E[Z]/x.Using the threshold in the numerator.
The inequality has the expectation in the numerator and the threshold in the denominator.
Fix:
Use the form P[Z ≥ x] ≤ E[Z]/x.Assuming a higher threshold can increase the Markov bound.
With a fixed expectation, increasing the denominator makes the ratio smaller.
Fix:
Recognize 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
- A nonnegative random variable never takes a value below zero.
- Expectation E[Z] represents the average value of the random variable.
- For a nonnegative random variable, Markov's inequality is P[Z ≥ x] ≤ E[Z]/x.
- The result is an upper bound, not necessarily the exact probability.
- With a fixed expectation, increasing the threshold makes the Markov bound smaller or equal.
Key Takeaways
- 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.