The random variable X is said to be stochastically larger...

The random variable $X$ is said to be stochastically larger than the random variable $Y$ if $$ P(X>z) \geq P(Y>z) $$ for all real $z$, with strict inequality holding for at least one $z$ value. Show that this requires that the cdfs enjoy the following property: $$ F_{X}(z) \leq F_{Y}(z) $$ for all real $z$, with strict inequality holding for at least one $z$ value.

The random variable $X$ is said to be stochastically larger than the random variable $Y$ if
$$
P(X>z) \geq P(Y>z)
$$
for all real $z$, with strict inequality holding for at least one $z$ value. Show that this requires that the cdfs enjoy the following property:
$$
F_{X}(z) \leq F_{Y}(z)
$$
for all real $z$, with strict inequality holding for at least one $z$ value.

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Key Concepts

Stochastic Ordering

Stochastic ordering is a method of comparing random variables based on the likelihood that one variable produces larger values than the other. It establishes a formal sense in which one random variable is 'larger' than another by comparing their tail probabilities across all thresholds.

Cumulative Distribution Function (CDF)

The CDF of a random variable gives the probability that the variable will take a value less than or equal to a given threshold. It is a fundamental tool in probability theory used to fully characterize the distribution of a random variable, impacting methods of comparison between different random variables.

Survival Function

The survival function, which is the complement of the CDF, provides the probability that the variable exceeds a certain value. It is central to the concept of stochastic ordering, as comparing the survival functions directly leads to conclusions about the ordering of the CDFs.

Complement Rule in Probability

The complement rule in probability states that the probability of an event not occurring is equal to one minus the probability that it does occur. This relationship between a random variable's CDF and its survival function is essential in demonstrating that a higher survival probability implies a lower CDF value.

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