**What is the formula for expected value?**
Expected value, also known as the mean or average, is a concept frequently utilized in statistics and probability theory. It represents the long-term average outcome of a random variable or event. The formula for expected value is as follows:
**Expected Value (E) = Σ (X * P(X))**
In this formula, “X” represents the possible outcomes of the random variable, and “P(X)” represents the probability of each outcome occurring. The expected value is obtained by multiplying each possible outcome by its corresponding probability, summing up these values, and obtaining the final result.
1. What is a random variable?
A random variable is a variable whose possible values depend on the outcome of a random event.
2. What does the expected value indicate?
The expected value provides an estimation of the average outcome of a random variable over the long run.
3. How is the expected value calculated for discrete random variables?
In the case of discrete random variables, the expected value is calculated using the formula mentioned above.
4. Can the expected value be negative?
Yes, the expected value can be negative if the random variable has negative outcomes.
5. How is the expected value calculated for continuous random variables?
For continuous random variables, the expected value is calculated using the integral of the value multiplied by its probability density function (pdf).
6. How is the expected value useful in decision-making?
The expected value helps decision-makers assess potential outcomes and make optimal choices based on their probabilities and associated values.
7. What is the relationship between expected value and variance?
Expected value and variance are both measures used to describe the behavior of random variables, but they represent different aspects. The expected value measures the central tendency, while the variance measures the dispersion or spread around the expected value.
8. Does the expected value guarantee a specific outcome?
No, the expected value does not guarantee a specific outcome. It represents the average outcome over a large number of trials or occurrences.
9. How is expected value applied in gambling?
In gambling, expected value is utilized to determine the potential profitability of different bets or games. Positive expected value indicates a favorable outcome in the long run.
10. Can expected value be used to predict individual outcomes?
Expected value is a tool for assessing long-term averages and probabilities, not for predicting specific outcomes in a single event.
11. What is the difference between expected value and expected utility?
Expected value focuses on the numerical average of a random variable, while expected utility considers the satisfaction or value associated with different outcomes.
12. Are there any limitations to using expected value?
Expected value assumes perfect knowledge of all probabilities, which may not always be accurate in real-world scenarios. It also does not account for potential outliers or extreme events that may significantly impact outcomes.