Concepts / Expectation of a Random Variable

Expectation of a Random Variable

Markov's inequality applies to nonnegative random variables.

  • Programming

From Average to Probability

Suppose Z is a random variable that never takes a negative value. You know its expectation, E[Z], but you want to control how likely Z is to reach or exceed a chosen threshold x. Markov's inequality connects these two pieces of information: it uses the expectation to produce an upper bound on the probability of the event Z ≥ x.

The result is an upper bound, not necessarily the exact probability.

What Expectation Tells You

The expectation of a random variable describes its average value. It combines the possible values of the variable with their probabilities to produce a weighted average. In Markov's inequality, the expectation E[Z] supplies the average-size information used to control the probability of reaching a threshold.

weighted togetherweighted togetherPossible valuesValues of ZE[Z]Average valueProbabilitiesLikelihoods of values
How do possible values and their probabilities combine to determine the average value?

The Nonnegative Requirement

A nonnegative random variable is a random variable whose values are always at least zero. Markov's inequality applies only under this condition.

The condition matters because Markov's inequality is stated for a variable Z whose values are not negative, together with its expectation E[Z] and a threshold x. The event being bounded is Z ≥ x: Z reaches or exceeds the threshold.

Markov's Inequality

P[Z ≥ x] ≤ E[Z] / x

To use the inequality, identify three ingredients: the nonnegative random variable Z, its expectation E[Z], and the chosen threshold x. Divide the expectation by the threshold. The resulting ratio is an upper bound for the probability that Z reaches or exceeds x.

allows use ofwithnumeratordenominatorboundsZ ≥ 0Required conditionE[Z]ExpectationE[Z] / xUpper-bound ratioP[Z ≥ x]At most E[Z] / xxChosen threshold
How does Markov's inequality use nonnegativity, the expectation, and a threshold to produce an upper bound?

Substituting Known Values

Bounding a Threshold Event

Suppose Z is nonnegative, E[Z] = 6, and the threshold is x = 15. Use Markov's inequality to bound P[Z ≥ 15].

Identify the inequality: For a nonnegative random variable, use P[Z ≥ x] ≤ E[Z] / x.

Substitute the expectation and threshold: Replace E[Z] with 6 and x with 15, giving P[Z ≥ 15] ≤ 6 / 15.

Evaluate the ratio: The ratio 6 / 15 equals 0.4.

P[Z ≥ 15] ≤ 0.4. The probability is at most 0.4; it is not necessarily equal to 0.4.

numeratordenominatorgives upper boundE[Z] = 6Known expectationx = 15Chosen threshold6 / 150.4P[Z ≥ 15]≤ 0.4
What changes when the expectation and threshold are substituted into Markov's inequality?
Output
P[Z ≥ 15] ≤ 6 / 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. When x increases, the denominator becomes larger, so the ratio becomes smaller. The probability of reaching or exceeding the higher threshold therefore has a no-larger Markov bound. The probability P[Z ≥ x] is also monotonically nonincreasing as the threshold increases.

divide expectation bydivide expectation byxOriginal thresholdE[Z] / xOriginal boundlarger xHigher thresholdE[Z] / larger xSmaller bound
How does increasing the threshold change the probability bound when the expectation stays fixed?

Common Mistakes

  • Applying Markov's inequality without checking nonnegativity.

    Markov's inequality requires the random variable to never take a negative value.

    Fix: First verify that Z is nonnegative.

  • Treating the upper bound as the exact probability.

    The inequality states that the probability is at most 0.4.

    Fix: Write P[Z ≥ 15] ≤ 0.4 and interpret it as an upper bound.

  • Using the wrong threshold event.

    Markov's inequality specifically bounds the event that Z reaches or exceeds x.

    Fix: Match the event and threshold carefully: P[Z ≥ x].

  • Expecting a higher threshold to produce a larger bound.

    With E[Z] fixed, increasing the denominator makes the ratio smaller.

    Fix: Recalculate E[Z] / x and compare the two ratios.

Try the Bound

EASY

A nonnegative random variable Z has expectation E[Z] = 6. Use Markov's inequality to write an upper bound for the probability that Z reaches or exceeds a chosen threshold x. Then explain what happens to the bound if x increases while E[Z] remains fixed.

Hints
  • Start with P[Z ≥ x] ≤ E[Z] / x.
  • Substitute E[Z] = 6.
  • Compare 6 / x with 6 divided by a larger threshold.

What do you think happens?

If E[Z] stays fixed and the threshold increases, does the Markov bound become larger, stay the same, or become smaller?

  • Larger
  • The same
  • Smaller
Reveal answer

Answer: Smaller

The bound is E[Z] / x. Increasing x increases the denominator, so the ratio becomes smaller.

Key Takeaways

  1. A nonnegative random variable never takes a negative value.
  2. Expectation E[Z] represents the average value used in Markov's inequality.
  3. For a nonnegative random variable, P[Z ≥ x] ≤ E[Z] / x.
  4. The result is an upper bound, not necessarily the exact probability.
  5. With a fixed expectation, increasing the threshold makes the Markov bound smaller.

Key Takeaways

  • A nonnegative random variable has values that are always at least zero.
  • Expectation describes the average value of a random variable.
  • Markov's inequality connects E[Z] with the threshold event P[Z ≥ x].
  • The bound P[Z ≥ x] ≤ E[Z] / x gives an upper limit, not necessarily the exact probability.
  • For fixed expectation, a higher threshold produces a smaller probability bound.