Understanding the Random Variable
A random variable is a numerical outcome of a random process. In this case, we have a variable x that assumes the same value multiple times.
Values and Probabilities
- The random variable x takes the value x (the same value) four times.
- The corresponding probabilities are P, P, P, and P.
Calculating the Expected Value
The expected value (E) of a discrete random variable is calculated using the formula:
E(X) = Σ (x_i * P_i)
Where:
- x_i represents the values the random variable can take.
- P_i represents the probabilities associated with those values.
In this scenario:
- Values: x, x, x, x
- Probabilities: P, P, P, P
Expected Value Calculation Steps
1. Substituting Values:
- E(X) = x * P + x * P + x * P + x * P
2. Simplifying:
- Combine like terms:
- E(X) = 4x * P
Conclusion
The expected value of the random variable x, which takes the value x with a probability P, summed over four instances, is given by:
- E(X) = 4x * P
This means that the expected outcome, when considering the probabilities, is simply four times the product of the value x and its probability P. This concept is foundational in probability theory and statistics, providing insights into the average outcome of random variables over time.