The expected value (often denoted as $E(X)$) is a fundamental concept in probability and statistics that represents the average or mean value of a random variable if the experiment or process were repeated infinitely many times.

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How to Compute Expected Value (E(X)):

The method of calculating the expected value depends on whether the random variable is discrete or continuous.

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1. For a Discrete Random Variable:

A discrete random variable takes on a finite or countable number of values. Its expected value is computed as: $E(X) = \sum_{i} x_i \cdot P(x_i)$ Where:

• $x_i$: the $i$-th possible value of the random variable $X$,
• $P(x_i)$: the probability of $X = x_i$,
• The summation runs over all possible values of $X$.

Example:

Suppose a die is rolled, and $X$ is the outcome. The expected value is: $E(X) = \sum_{i=1}^{6} i \cdot P(i) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6} = 3.5$

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2. For a Continuous Random Variable:

A continuous random variable can take on infinitely many values. Its expected value is computed as: $E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$ Where:

• $f(x)$: the probability density function (PDF) of $X$,
• The integral is over all possible values of $X$.

Example:

If $X$ has a uniform distribution between 0 and 1 ($f(x) = 1$ for $0 \leq x \leq 1$), then: $E(X) = \int_{0}^{1} x \cdot 1 \, dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1}{2}$

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Why is Expected Value Important?

Expected value is a central measure in decision-making, economics, and statistics. It helps:

• Predict long-term averages in stochastic processes.
• Evaluate fairness or profitability in games of chance or investments.
• Analyze outcomes in risk assessments.

Would you like a deeper explanation, a detailed example, or to clarify a specific step?

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Expanding Questions:

1. How does the expected value differ between discrete and continuous random variables?
2. Can expected values be negative? If so, when does this occur?
3. How is the concept of expected value applied in real-world scenarios like insurance or gambling?
4. What happens if the probabilities in a discrete random variable do not sum to 1?
5. How do we compute the expected value of a random variable when given a cumulative distribution function (CDF)?

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Tip:

Always check that the sum of probabilities (discrete case) or the integral of the PDF (continuous case) equals 1 before calculating the expected value.