Step 1: Understanding the Concept:
In probability theory and statistics, the "Mean" or "Expected Value" (denoted as \( \mu \) or \( E[X] \)) of a discrete random variable provides a measure of the central tendency of the distribution.
It represents the long-run average outcome if a random experiment were to be repeated many times.
Unlike a simple arithmetic mean, which treats all outcomes equally, the expected value is a "weighted average."
Each possible outcome value is weighted by its probability of occurring. Outcomes with higher probabilities have a greater influence on the final mean.
Step 2: Key Formula or Approach:
The formula for calculating the mean of a discrete random variable is:
\[ \mu = \sum [x \cdot P(x)] \]
This involves multiplying each value \( x \) in the distribution table by its corresponding probability \( P(x) \) and then summing all these individual products.
Step 3: Detailed Explanation:
Let's extract the pairs from the provided distribution table:
- For \( X = 1 \), \( P(1) = 0.2 \)
- For \( X = 2 \), \( P(2) = 0.5 \)
- For \( X = 3 \), \( P(3) = 0.3 \)
Now, calculate the product for each pair:
\[ \text{Product 1} = 1 \times 0.2 = 0.2 \]
\[ \text{Product 2} = 2 \times 0.5 = 1.0 \]
\[ \text{Product 3} = 3 \times 0.3 = 0.9 \]
To find the final expected value, add these three products together:
\[ E[X] = 0.2 + 1.0 + 0.9 \]
\[ E[X] = 1.2 + 0.9 \]
\[ E[X] = 2.1 \]
The value 2.1 tells us that while 2.1 itself is not a possible outcome (the outcomes are only 1, 2, or 3), it represents the center of the distribution's weight.
Step 4: Final Answer:
The mean (expected value) of the random variable \( X \) is 2.1.