Question

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.

Short answer

The original inequality can be rewritten in terms of cumulative distribution functions, resulting in \(F_{X}(z) \leq F_{Y}(z)\) which is the property mentioned in the exercise. This holds for all real \(z\), with strict inequality present for at least one \(z\) value.

Step-by-step solution

Foundation on terms

Firstly, 'Stochastically larger' means that the probability of one random variable (in this case, \(X\)) being larger than an arbitrary value \(z\) is greater than or equal to the probability of another random variable (here, \(Y\)) being larger than the same value \(z\) for all real \(z\). This has to be true with strict inequality holding for at least one value of \(z\). The 'Cumulative Distribution Function' (CDF) \(F(z)\) of a random variable gives the probability that the variable takes a value less than or equal to \(z\).

Redefine in terms of CDFs

Since the definition of stochastic dominance is given in terms of values larger than \(z\), we can redefine these probabilities in terms of cumulative distribution functions. This works because \(P(X>z) = 1 - P(X \leq z)\) and \(P(Y>z) = 1 - P(Y\leq z)\) respectively.

Compare the equations

Now, we compare these redefined probabilities. \(1 - F_{X}(z) \geq 1 - F_{Y}(z)\), which we can rearrange to \(F_{X}(z) \leq F_{Y}(z)\). This holds for all real \(z\) with strict inequality for at least one \(z\) value, meaning the cdfs do indeed possess the described property.

Key concepts

Random Variables

A random variable is a fundamental concept in probability theory. It assigns a numerical outcome to each event in a sample space, making abstract concepts more tangible. Random variables can be either discrete or continuous.

• A discrete random variable has a finite or countably infinite set of outcomes. For example, the number of heads in three flips of a coin.
• A continuous random variable has an infinite set of possible outcomes, often corresponding to measurements. An example is the height of students in a class.

Random variables are usually denoted by capital letters like \(X,Y,Z\). They serve as the basis for further concepts in probability and statistics, such as probability distributions and cumulative distribution functions (CDFs). Understanding random variables helps in modeling real-world processes, predicting outcomes, and making informed decisions based on data analysis. Each outcome's probability underpins how we interpret and utilize these variables.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a useful tool associated with a random variable. It describes the probability that the random variable will take a value less than or equal to a certain point.The formula for a CDF, \(F_X(z)\), is expressed as:\[ F_X(z) = P(X \leq z) \]Key properties of the CDF include:

• Non-decreasing: As \(z\) increases, \(F(z)\) either stays the same or increases, reflecting higher cumulative probability.
• Right-continuous: The function does not jump for any value \(z\), offering a smooth probability progression.
• Boundaries: \(F(+\infty) = 1\) and \(F(-\infty) = 0\), indicating that the probability spans from 0 to 1 everywhere.

CDFs are particularly useful when comparing random variables. They give insights into how probabilities distribute across different values, offering a broader understanding than just a probability mass or density function alone.

Inequality in Probability

In probability, inequalities form the basis for describing relationships between random variables' outcomes. One common inequality involves stochastic dominance, where one variable is said to dominate another if its outcomes have consistently higher probabilities.

• Stochastic dominance exists when, for all values \(z\), the probability \(P(X > z)\) is greater than or equal to \(P(Y > z)\).
• It's explicitly expressed as \(F_X(z) \leq F_Y(z)\) for all \(z\), meaning the CDF of \(X\) is less than or equal to the CDF of \(Y\).
• For stochastic dominance to hold, there must be at least one value of \(z\) where the inequality is strict.

These inequalities are crucial in decision-making and economics, as they offer a way to compare different options without exact numerical probabilities. It effectively provides a hierarchy or ranking of random variables based on their distribution.