(8) Let X be an r.v. whose possible values are 0,1,2, …, with CDF F. In some countries, rather t

step 1 We have here a random variable x whose cumulative distribution function is given by this express

(8) Let X be an r.v. whose possible values are 0,1,2, …,...

(8) Let $X$ be an r.v. whose possible values are $0,1,2, \ldots$, with CDF $F$. In some countries, rather than using a CDF, the convention is to use the function $G$ defined by $G(x)=$ $P(X<x)$ to specify a distribution. Find a way to convert from $F$ to $G$, i.e., if $F$ is a known function, show how to obtain $G(x)$ for all real $x$.

Key Concepts

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function, often denoted as F(x), specifies the probability that a random variable X takes a value less than or equal to x. For discrete random variables, it is a step function that increases only at the possible values of the variable. Understanding the CDF is fundamental in probability theory, as it provides a complete description of the distribution of the random variable.

Left-Hand Limit of the CDF

For discrete random variables, the probability of the random variable being less than a given value x, denoted as G(x) = P(X < x), can be obtained by taking the left-hand limit of the CDF at x. This is because the CDF F(x) is right-continuous, meaning that its value at x includes the probability at x itself. Therefore, the conversion from F(x) to the function G(x) is achieved by G(x) = lim(y -> x-) F(y). This concept is crucial when distinguishing between probabilities less than a value and those less than or equal to a value.

Discrete Random Variable

A discrete random variable is characterized by having a countable set of possible outcomes, such as 0, 1, 2, and so on. In the context of converting CDFs, its discrete nature implies that the CDF is a step function, making the distinction between the probability at a point and the probability of being less than the point significant. Understanding the properties of discrete random variables allows one to appropriately manipulate and transform the distribution functions according to different conventions.

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