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In probability theory, a **discrete probability distribution** is one that shows the probabilities of outcomes of a discrete random variable. A discrete random variable is one that has a countable number of distinct values. For example, the variable M in the exercise can take on the values 0, 1, or 2. Each of these values has a corresponding probability: \( P(M=0)=0.2 \), \( P(M=1)=0.5 \), and \( P(M=2)=0.3 \). These probabilities must sum up to 1, since one of these outcomes must occur. This means: \begin{{equation}} P(M=0) + P(M=1) + P(M=2) = 1 \rightarrow 0.2 + 0.5 + 0.3 = 1 \text{.} Here's a convenient way to look at discrete probabilities:

• Probabilities are assigned to specific, separate outcomes.
• The sum of all probabilities in the distribution is always 1.
• The probabilities can only take non-negative values.

Understanding these basic rules helps you work with any discrete probability distribution more effectively.

A **histogram** is a visual representation of the frequency or probability of different outcomes in a data set. When dealing with probability distributions, histograms help us see how probabilities are distributed across different values. In our exercise, we need to draw a histogram for the random variable M, where:

• For M=0, the probability is 0.2
• For M=1, the probability is 0.5
• For M=2, the probability is 0.3

To create the histogram:

• Draw a horizontal line (x-axis) and mark the values 0, 1, and 2.
• Draw a vertical line (y-axis) and mark it with suitable probabilities, at least up to 0.5.
• For each value on the x-axis, draw a vertical bar reaching up to its corresponding probability on the y-axis.

This visual allows us to quickly see which outcomes are most and least likely. The higher the bar, the higher the probability of that outcome. Labeling the x-axis as 'Values of M' and the y-axis as 'Probability' further clarifies the representation.

A **random variable** is a variable that takes on numerical values, determined by the outcome of a random phenomenon. Random variables can be classified into two types: discrete and continuous. In our exercise, M is a **discrete random variable** because it has specific, countable outcomes (0, 1, and 2). Here are some key traits of random variables:

• They are used to quantify outcomes of random phenomena.
• They have associated probabilities for each outcome.
• Discrete random variables take on countable values, while continuous random variables take on an infinite number of possible values within a range.

Understanding random variables helps you model and analyze real-world random processes. They are core elements of probability theory and statistics, making them essential for data science, finance, engineering, and many other fields.