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A probability distribution is a statistical function that describes the likelihood of obtaining the possible values that a random variable can assume. Each of these possible values is associated with a probability, which indicates how likely it is for this value to occur. In the context of a discrete random variable, like in this exercise, the distribution is discrete, meaning the outcomes are distinct and finite.

• The sum of all probabilities in a distribution must be 1, ensuring that one of the possible outcomes will certainly occur.
• Each probability value must be between 0 and 1, indicating the range within which each event can occur with certainty or impossibility.

In the example provided, we have a discrete probability distribution:

• Value of X: 0 with Probability: 0.5
• Value of X: 1 with Probability: 0.2
• Value of X: 2 with Probability: 0.2
• Value of X: 3 with Probability: 0.1

Adding all probabilities together will give a total of 1, showing a full distribution. Understanding this framework is crucial because it sets the stage for calculating other metrics, such as the expected value.

A discrete random variable is a type of random variable that can take on a finite or countably infinite set of values. These types of variables are common in probability distributions where outcomes are distinct, such as rolling a die or flipping a coin. Here are some key characteristics of discrete random variables:

• They only take specific, separate values. For example, in this exercise, the values are 0, 1, 2, and 3, and nothing in between.
• Associated with each value is a certain probability, as part of the probability distribution.
• Discrete random variables are often easier to handle mathematically because they have defined, distinct outcomes.

For the given problem, the variable X is discrete. It represents distinct outcomes each with its probability, making it an excellent example to understand how we use these discrete values to compute further statistical measures like expected value.

Expected value is a fundamental concept in probability that provides a measure of the central tendency of a random variable. It's like the weighted average of all possible values of the random variable, where the weights are the probabilities of each value occurring.To calculate the expected value for a discrete random variable, you use the formula:\[E(X) = \sum_{i=1}^n x_i \cdot P(X = x_i)\]Here's what each term means:

• \(x_i\) is a possible value of the random variable X.
• \(P(X = x_i)\) is the probability that X takes the value \(x_i\).
• \(\sum\) indicates that you need to add all these products together for all values \(x_i\) that X can take.

For our exercise,

• The expected value is calculated by summing the products of each value and its corresponding probability: \(E(X) = 0 \cdot 0.5 + 1 \cdot 0.2 + 2 \cdot 0.2 + 3 \cdot 0.1\)
• This yields \(E(X) = 0 + 0.2 + 0.4 + 0.3 = 0.9\).

Thus, the expected value of X is 0.9. This concept helps to predict the average outcome if the experiment is repeated many times, providing substantial insight into the behavior of the random variable.