Final answer:
The question explores the concept of a random variable X, its probability distribution function, and cumulative distribution function in mathematics. It involves understanding how probabilities are assigned to different outcomes of X and how these probabilities accumulate over a range of values.
Step-by-step explanation:
The question discusses the characteristics of a random variable X, including its distribution, values it may take on, its probability density function (PDF), and the cumulative distribution function (CDF).
Defining the Random Variable X:
A random variable X represents a numerical outcome of a random phenomenon. In one scenario, it might denote the amount of money a student carries. This random variable can be discrete or continuous, depending on the context. A discrete random variable, for example, may take on values like 0, 1, 2, ... in the case of counting something like the number of children a person has. A continuous random variable usually represents measurements and can take on any value within a given range.
Probability Distribution Function:
The probability distribution of a random variable X describes how probabilities are assigned to each possible value that X can take on. For instance, the exponential distribution is defined by the PDF f(x) = me−mx, where m is the rate parameter and e is the base of the natural logarithm. When m = 0.25, this defines the exponential distribution as X~ Exp(0.25), indicating that the mean (μ) and standard deviation (σ) are equal and take the value 4 (∵ 1/m = μ = σ).
Cumulative Distribution Function:
The cumulative distribution function CDF of a random variable X, represented as P(X ≤ x), gives the probability that X will take a value less than or equal to x. It is a function that increases from 0 to 1 as x moves through the values in the domain of X.