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A discrete random variable is a type of random variable that takes on a countable number of distinct values. These values are often whole numbers, and each of these values corresponds to a specific probability. This means that the random variable can only be one of these fixed values at any given time.
Consider a simple example of rolling a six-sided die. The possible outcomes when rolling the die are 1, 2, 3, 4, 5, and 6. Hence, the roll outcome is a discrete random variable. Each roll outcome can be associated with a probability (in this case, typically 1/6 if the die is fair).
Key characteristics of discrete random variables include:

• The outcomes can be listed, like 1, 2, 3, etc.
• Each outcome has a probability that sums up to 1 when considered for all possible outcomes.
• The variable doesn’t assume any value between those in the listing; no fractional values exist between whole number outcomes.

In the problem at hand, the random variable \( Y \) can take the values of 1, 2, 3, or 4, each with a specified probability. Again, this highlights the nature of \( Y \) as a discrete random variable.

A probability distribution describes how probabilities are distributed over the values of the random variable. For discrete random variables, this is often represented in a probability mass function, where each possible value of the variable is assigned a probability.
Using our exercise, the probability distribution of \( Y \) was given with each value of \( y \) (1 through 4) having specific probabilities: 0.4, 0.3, 0.2, and 0.1 respectively. Summing these probabilities confirms they equal 1, ensuring it's a valid probability distribution.
Essential elements of a probability distribution include:

• Each possible outcome of the random variable has a unique probability associated with it.
• The probabilities must be non-negative.
• The total probability across all outcomes must sum to 1.

The probability distribution describes everything we need to understand about the potential outcomes of the random variable \( Y \). It allows us to calculate cumulative probabilities and other statistics about \( Y \).

Step functions are essential when discussing cumulative distribution functions of discrete variables. A step function is a piecewise constant function that jumps from one value to another. In our context, the cumulative distribution function \( F(y) \) is a step function.
For a discrete random variable, the cumulative distribution function \( F(y) \) represents the probability that the variable is less than or equal to \( y \). As \( y \) increases, \( F(y) \) jumps to higher values at each possible outcome of the variable. This creates a staircase-like pattern on the graph.
Features of step functions in CDFs include:

• Horizontal segments between the jumps where \( F(y) \) remains constant.
• A jump occurs at each possible value of the discrete random variable, where a new probability is accumulated.
• The final step reaches a total probability of 1.

In our exercise, \( F(y) \) jumps from 0 to 1, with distinctive steps at 1, 2, 3, and 4, illustrating how step functions depict the probability accumulation visually across possible values of \( Y \).