Expected Value of a Function of a Random Variable
Discrete Random Variables
Let X be a discrete random variable with PMF p(x). The expected value of a function g(x) of X is defined as:
E[g(X)] = Σ[g(x) * p(x)]
where the sum is taken over all possible values of x.Continuous Random Variables
For a continuous random variable X with PDF f(x), the expected value of g(x) is defined as:
E[g(X)] = ∫[-∞, ∞] g(x) * f(x) dx
Properties
The expected value of a function of a random variable has the following properties:
- E[a * g(X)] = a * E[g(X)], where a is a constant.
- E[g(X) + h(X)] = E[g(X)] + E[h(X)]
- E[g(h(X))] = E[g(Y)], where Y = h(X).
Applications
The expected value of a function of a random variable is widely used in probability theory and statistics. Some common applications include:
- Finding the mean of a random variable.
- Evaluating the expected return of a random investment.
- Calculating the probability of an event using the law of total probability.
Conclusion
The expected value of a function of a random variable is a fundamental concept in probability theory. It provides a way to summarize the probability distribution of a random variable and is used in a wide range of applications.
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