Worked Tail Comparisons for Discrete and Continuous Laws

What a Tail Comparison Tells Us

A comparison of two distributions is often most useful when stated in terms of thresholds: how likely is each random variable to exceed a given value? For random variables \(X\) and \(Y\), the usual stochastic order \(X\le_{\mathrm{st}}Y\) means that \(P(X>t)\le P(Y>t)\) for every real \(t\). Equivalently, their CDFs satisfy \(F_X(t)\ge F_Y(t)\) for every \(t\). This tutorial develops that comparison by calculation for Bernoulli, binomial, and exponential laws.

A tail comparison also predicts the direction of expectations of increasing functions, provided the expectations exist. The examples below make that connection concrete. They also show why checking a mean, a single threshold, or a single endpoint is not enough to establish an order.

Bernoulli Distributions: One Tail Determines the Comparison

If \(X\sim\operatorname{Bernoulli}(p)\), then \(P(X=1)=p\) and \(P(X=0)=1-p\), where \(0\le p\le1\). Its CDF and survival probability are

Thus, for two Bernoulli variables with success probabilities \(p_1\) and \(p_2\), the only interval on which their survival probabilities can differ is \(0\le t<1\). It follows directly that \(\operatorname{Bernoulli}(p_1)\le_{\mathrm{st}}\operatorname{Bernoulli}(p_2)\) exactly when \(p_1\le p_2\).

Binomial Distributions: Comparing Every Integer Threshold

For \(X\sim\operatorname{Binomial}(n,p)\), the probability mass at \(j\in\{0,\ldots,n\}\) is \(\binom{n}{j}p^j(1-p)^{n-j}\). Since \(X\) is integer-valued, its survival probability changes only at integer thresholds: for integer \(k\), \(P(X\ge k)\) is the tail at the threshold just below \(k\). Comparing these tails for \(k=1,\ldots,n\), together with the trivial tails outside the support, suffices to compare two binomial laws with the same \(n\).

The following derivative calculation verifies that, for fixed \(n\), every such tail is nondecreasing in \(p\). For \(1\le k\le n\), write

For \(0<p<1\), differentiate term by term; the sum is finite. Using \(j\binom{n}{j}=n\binom{n-1}{j-1}\) and \((n-j)\binom{n}{j}=n\binom{n-1}{j}\), the positive terms cancel the negative terms from adjacent indices, leaving

Continuity of \(S_k(p)\), which is a polynomial, extends the nondecreasing property to \(p=0\) and \(p=1\). Therefore, with \(n\) fixed and \(p_1\le p_2\), \(\operatorname{Binomial}(n,p_1)\le_{\mathrm{st}}\operatorname{Binomial}(n,p_2)\). For \(0<p_1<p_2<1\), the derivative is positive for every \(1\le k\le n\), so each of these integer-threshold tails is strictly larger at \(p_2\).

The qualification “with \(n\) fixed” matters. Equal means alone do not establish a stochastic order. For example, let \(U\sim\operatorname{Binomial}(1,0.5)\) and \(V\sim\operatorname{Binomial}(2,0.25)\). Both have mean \(0.5\), but

The tail inequalities go in opposite directions, so neither law is stochastically larger than the other. Each calculation is necessary: checking only one threshold, or only the mean, would miss this failure of ordering.

Increasing-Function Expectations from Integer Tails

For a nonnegative integer-valued random variable, tail probabilities also give a direct formula for expectations. Let \(g\) be nondecreasing on the nonnegative integers, with finite real values, and define \(\Delta g(k)=g(k)-g(k-1)\ge0\) for \(k\ge1\). For each nonnegative integer \(m\), telescoping gives \(g(m)=g(0)+\sum_{k=1}^{m}\Delta g(k)\). Applying this identity to \(X\) yields

Every summand after \(g(0)\) is nonnegative. Tonelli's theorem therefore permits interchanging the expectation and the infinite sum, even if the resulting expectation is infinite:

In particular, if \(X\le_{\mathrm{st}}Y\) and both are nonnegative integer-valued, each tail term for \(X\) is no larger than the corresponding term for \(Y\). The formula then gives \(E[g(X)]\le E[g(Y)]\) for every such nondecreasing \(g\), with the expectations understood as extended expectations when necessary. For a finite-support variable, the sum has only finitely many nonzero terms.

Exponential Distributions: Rate Changes and Continuous Tails

Let \(X\sim\operatorname{Exponential}(\lambda)\), where \(\lambda>0\) is the rate. Its density is \(f_X(x)=\lambda e^{-\lambda x}\) for \(x\ge0\), and zero otherwise. Integrating the density gives, for \(t\ge0\),

For \(t<0\), \(F_X(t)=0\) and \(P(X>t)=1\), because \(X\) is nonnegative. If \(0<\lambda_1<\lambda_2\), then \(e^{-\lambda_2t}\le e^{-\lambda_1t}\) for every \(t\ge0\). Equality holds at \(t=0\), since both survival probabilities are 1 there; the inequality is strict for every \(t>0\). For \(t<0\), both probabilities equal 1. Thus the larger rate has no larger survival probability at every nonnegative threshold, and a strictly smaller survival probability at every positive threshold. In stochastic-order notation, \(\operatorname{Exponential}(\lambda_2)\le_{\mathrm{st}}\operatorname{Exponential}(\lambda_1)\).

Common Mistakes and a Practical Check

• Calling the exponential inequality strict at zero. At \(t=0\), every positive-rate exponential law has survival probability 1. The comparison for rates \(\lambda_2>\lambda_1\) is non-strict at \(t=0\) and strict only for \(t>0\).
• Reversing the direction when comparing CDFs and tails. If \(X\le_{\mathrm{st}}Y\), then \(P(X>t)\le P(Y>t)\), but \(F_X(t)\ge F_Y(t)\). The CDF inequality points in the opposite direction.
• Inferring an order from means or one threshold. Equal means, or a favorable comparison at one cutoff, says nothing about all other cutoffs. The binomial pair with equal means above has tails crossing at \(k=1\) and \(k=2\).
• Using a function that is not increasing on the relevant support. The expectation comparison concerns increasing functions. For example, \(x^2\) is increasing on nonnegative values, but not on the whole real line.

For a discrete law, list the possible integer thresholds and calculate each tail; for an exponential law, use its survival formula and check the boundary \(t=0\) separately. After verifying the comparison at every threshold, expectations of selected increasing functions provide useful numerical checks, not substitutes for the tail calculation.