(a) Give an example of r.v.s X and Y such that FX(x) ≤FY(x) for all x, where the inequality is s

step 1 Problem 8 this is a cumulative distribution function for the random variable x We want to sketch

(a) Give an example of r.v.s X and Y such that FX(x) ≤FY(x)...

(a) Give an example of r.v.s $X$ and $Y$ such that $F_{X}(x) \leq F_{Y}(x)$ for all $x$, where the inequality is strict for some $x$. Here $F_{X}$ is the CDF of $X$ and $F_{Y}$ is the CDF of $Y$. For the example you gave, sketch the CDFs of both $X$ and $Y$ on the same axes. Then sketch their PMFs on a second set of axes. (b) In Part (a), you found an example of two different CDFs where the first is less than or equal to the second everywhere. Is it possible to find two different PMFs where the first is less than or equal to the second everywhere? In other words, find discrete r.v.s $X$ and $Y$ such that $P(X=x) \leq P(Y=x)$ for all $x$, where the inequality is strict for some $x$, or show that it is impossible to find such r.v.s.

(a) Give an example of r.v.s $X$ and $Y$ such that $F_{X}(x) \leq F_{Y}(x)$ for all $x$, where the inequality is strict for some $x$. Here $F_{X}$ is the CDF of $X$ and $F_{Y}$ is the CDF of $Y$. For the example you gave, sketch the CDFs of both $X$ and $Y$ on the same axes. Then sketch their PMFs on a second set of axes.
(b) In Part (a), you found an example of two different CDFs where the first is less than or equal to the second everywhere. Is it possible to find two different PMFs where the first is less than or equal to the second everywhere? In other words, find discrete r.v.s $X$ and $Y$ such that $P(X=x) \leq P(Y=x)$ for all $x$, where the inequality is strict for some $x$, or show that it is impossible to find such r.v.s.

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Key Concepts

Graphical Representation of Distributions

Graphing the CDFs and PMFs offers a visual tool for understanding and comparing probability distributions. Sketching these functions helps to easily identify differences in the behavior of the distributions, such as shifts in mass or differences in cumulative probabilities.

Stochastic Ordering

Stochastic ordering is a method for comparing random variables by their distributions. A common form, first order stochastic dominance, occurs when the CDF of one random variable is always less than or equal to that of another, implying that it tends to yield higher values. This concept is key when assessing inequalities between CDFs or PMFs.

Cumulative Distribution Function (CDF)

A CDF is a function that maps any value x to the probability that a random variable takes on a value less than or equal to x. It serves as a complete description of the distribution of a random variable, allowing comparisons between different distributions by looking at their accumulated probabilities.

Probability Mass Function (PMF)

For discrete random variables, a PMF assigns the probability that the variable is equal to a specific value. It provides a detailed, point-by-point view of a distribution and is useful for calculating exact probabilities and performing comparisons between different discrete distributions.

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