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The expected value is like the weighted average of all possible outcomes of a random variable. It tells us the central or "average" value that you would expect from an experiment if it is repeated many times.
When calculating the expected value, it is important to understand that it doesn't always correspond to one of the actual outcomes the variable can take.
In mathematical terms, if a random variable \( x \) can take values \( x_i \) with probabilities \( P(x_i) \), then the expected value, \( E(x) \), is given by

• \( E(x) = \sum x_i P(x_i) \)

The calculation involves multiplying each possible value by its respective probability, then summing those results.
This helps to form an understanding of the long-term average if the same random process is conducted repeatedly.
A common misconception is that the expected value has to be one of the values that \( x \) can take, but examples show it can be a value not included in the original set of outcomes.

A probability distribution maps out all possible values of a random variable and their associated probabilities. It's like a complete description of the chances for each possible outcome.
This distribution is crucial to calculating concepts like expected value.
To imagine this, picture a list where each possible value is paired with a probability that tells you how likely that value is to occur.
Here's how it works:

• Every value the variable can take has a probability.
• All these probabilities add up to 1, symbolizing that one of these outcomes must happen.
• Examples of types of probability distributions include discrete and continuous distributions.

Discrete probability distributions concern outcomes you can actually list, like rolling a die.
Every side of the die (1 through 6) has a probability of 1/6.
This is a straightforward, tangible picture of possible outcomes.

A random variable represents a set of outcomes from a certain random process or experiment, often signified by a symbol like \( x \).
It is a way to map real-world scenarios into mathematical terms that statisticians can analyze using probability distributions.
Unlike a traditional variable, a random variable doesn't have a fixed value until the outcome of the experiment is known.
This means each time you perform the experiment, the variable can take on different values that are governed by probabilities.

• There are two primary types of random variables: discrete and continuous.
• Discrete random variables take on a finite number of values, like tossing a coin.
• Continuous random variables can take any value within a given range, like the height of people.

Understanding random variables is key to grasping probability and statistical concepts, as they are foundational elements that underline probability distributions and expected values.