Data Analysis and Probability Distributions

In probability, a probability distribution shows all the possible outcomes of a random event and the probability associated with each outcome. Understanding these distributions helps us analyze data, predict long-term results, and make informed decisions.

Rules of a Probability Distribution

For any discrete probability distribution (where outcomes can be counted), two main rules must always apply:

1. The probability of each individual outcome must be between $0$ and $1$: $0 \leq P(x) \leq 1$
2. The sum of all probabilities in the distribution must equal exactly $1$: $\sum P(x) = 1$

Expected Value $E(X)$

The expected value, denoted as $E(X)$, represents the long-run average outcome if you were to repeat an experiment many times. It is not necessarily an outcome that will actually happen in a single trial, but rather the "weighted average" of all possible outcomes.

The formula for expected value is:

$E(X) = \sum [x \cdot P(x)]$

This means you multiply each outcome $x$ by its probability $P(x)$, and then add all those products together.

Example 1: Verifying a Distribution and Finding $E(X)$

Imagine a random variable $X$ with the following distribution:

• $P(X=1) = 0.2$
• $P(X=2) = 0.5$
• $P(X=3) = 0.3$

Step 1: Verify it is a valid distribution. Check the sum of the probabilities: $0.2 + 0.5 + 0.3 = 1.0$ Since the sum is exactly $1$, this is a valid probability distribution.

Step 2: Find the expected value $E(X)$. Use the expected value formula: $E(X) = (1 \cdot 0.2) + (2 \cdot 0.5) + (3 \cdot 0.3)$ $E(X) = 0.2 + 1.0 + 0.9 = 2.1$

The long-run average outcome is $2.1$.

Example 2: Making Decisions with Expected Value

Expected value is highly useful in assessing risks, such as whether a game is worth playing.

Problem: A game costs \5$to play. You win$$20$with a probability of$0.2$, and you win$$0$ otherwise. What is the expected value of your net profit?

Step 1: Determine the net outcomes and their probabilities.

• Winning: If you win \20$, your *net profit* is$$20 - $5 = $15$. The probability is$0.2$.
• Losing: If you win \0$, your *net profit* is$$0 - $5 = -$5$. The probability is$1 - 0.2 = 0.8$.

Step 2: Calculate $E(X)$. $E(X) = (15 \cdot 0.2) + (-5 \cdot 0.8)$ $E(X) = 3 + (-4) = -1$

Conclusion: The expected value is -\1$. This means that, on average, you will lose$$1$ every time you play this game. Therefore, mathematically, it is not a favorable game to play in the long run.